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A H istor y of the A xi omat ic F ormulation

of Pr obabi l i ty f r om Bor el to Kolmogorov" Par t I

J A C K B A R O N E & A L B E R T N O V I K O F F

Communicated by M. KLINE

Abstract

This paper , the f i rs t o f two, traces the or ig ins of the m od ern ax iom at ic

formula t ion of P robabi l i ty Theory , which was f i r s t g iven in de f in i t ive form by

KOLMOGOROV in 1933. Ev en b efore tha t t ime, howe ver , a seq uenc e of develop -

ments , in i t i a t ed by a l an dm ark pape r of E . BOREL, were g iv ing r is e to prob lems ,theorems , and re formula t ions tha t inc reas ingly re la ted probabi l i ty to measure

theo ry and , in pa r t i cu la r , c l a r i fi ed the key ro le of countab le ad di t iv i ty in

P r oba b i l i t y T he o r y .

T h i s p a pe r de s cr i be s t he de ve l opme n t s f r om B O R E L ' S w or k t h r ough

F. HAUSDORFF'S. The major accomplishments of the per iod were BOREL's Zero-

On e L aw (also kno wn as the BOREL-CANTELLI Lem mas) , his St ro ng L aw of La rge

Nu mb ers , a nd h i s Con t inue d Fra c t ion The orem . W hat i s new is a de ta i l ed ana lys i s

of BOREL's or ig ina l p roofs , f rom w hich we t ry to acco unt for the root s

(ps yc ho log ica l a s we l l a s mathe mat ica l ) o f the man y f laws and inade quac ies in

BOREL's reasoning . We a l so document the inc reas ing rea l i za t ion of the l inkbe tween the theor ies of measu re and of prob abi l i ty in the pe r iod f ro m G . FABER to

F. HAUSDORFF. We indicate the mis leading emphasis given to independence as a

bas ic co nc ep t by BOREL and his equal ly un for tun ate assoc ia t ion of a HEINE-BOREL

lem ma wi th c oun tab le addi t iv i ty . Also or ig ina l is the (poss ib le ) genesi s we propo se

for each of the two examples cho sen by BOREL to exhib i t h i s new th eory ; in each

case we c it e a now neglec ted precurso r o f BOREL, one of them sure ly kno wn to

BOREL, the o the r , p rob ably so. The b r i e f ske tch of ins tances of the "CANTELLI"

lemma before CANTELLI 's publ icat ion is a lso or iginal .

W e descr ibe the interes t ing pole mic b etwe en F. BERNSTEIN an d BOREL

c onc e r n i ng t he C on t i nue d F r a c t i on T he o r e m , w h i c h se rve s as a r a r e i n st a nc e o f aco nt em po ra ry cr i t ic ism of BOREL's reasoning. W e also discuss HAUSDORFF'S pro of

of BOREL'S S t rong Law (which seems to be the f i rs t va l id pr oo f of the th eor em a long

the l ines sketched by BOREL).

I n r e t r o s pe ct , one ma y a s k w hy p r ob l e ms o f " ge o me t r i c " ( o r " c on t i nuo us " )

pro bab i l i ty did n ot give r ise to th e (KOLMOGOROV) view of pro ba bi l i ty as a form of

124 J. BARONE & A. NOVIKOFF

measu re , r a the r t han t he s tudy o f r epea t e d i ndepen den t t r i a ls , wh ich was BOREL 'S

a p p r o a c h . T h i s p a p e r s h o w s t h a t q u e s t i o n s o f " g e o m e t r i c " p r o b a b i l i t y w e r e a l w a y s

t h e e s s e n t i a l g u i d e t o t h e e a r l y de v e l o p m e n t o f t h e t h e o r y , d e s p i t e t h e c o n t r a r y

v i ewpo in t exh ib i t ed by BOREL 'S p re f e r r ed i n t e rp re t a t i on o f h i s own r e su l t s .

Table of Contents

0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

1. Brief Sketch of Major Results of BOREL (1909) . . . . . . . . . . . . . . . . . . . 126

2. "Countable Independence" as a Key Principle . . . . . . . . . . . . . . . . . . . 129

3. Denumerable Probability versus Measure Theory . . . . . . . . . . . . . . . . . . t31

4. The Evidence from BOREL'S Chapter I . . . . . . . . . . . . . . . . . . . . . . . 133

4.1. Denumerab le P robabi li ty Con tr as ted with Con tinuous P robabi li ty . . . . . . . . . 133

4.2. The Calculation of A 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

4.3. The Calculation of A1, Az, and A k . . . . . . . . . . . . . . . . . . . . . . . 135

4.4. The Calculation of A~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

4.5. Addi tio nal Evidence of the Pr im ac y o f C ou nt ab le Ind epe nden ce . . . . . . . . . . 139

4.6. S um ma ry o f BOREL'S C onc ept ual S ho rt co mi ng s fr om C ha pt er I . . . . . . . . . . 141

5. BOREL'S Chapter II: The Strong Law of Large Numbers . . . . . . . . . . . . . . . 142

5.1. The Setting of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 142

5.2. The Special Case Pn =½ for Dyadic Expansions . . . . . . . . . . . . . . . . . 143

5.3. Normal Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

5.4. A Possible Clue to the Genesis of BOREL'S Strong Law . . . . . . . . . . . . . . 147

6. BOREL'S Chapter III: Continued Fractions . . . . . . . . . . . . . . . . . . . . . 148

6.1. The Setting of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 148

6.2. The Derivati on of the Key Inequali ty; BOREL'S Cont inue d Fracti on The ore m . . . 149

6.3. Defects in Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

6.4. A Possible Clue to the Genesis of BOREL'S Continu ed Fraction Theor em . . . . .. 153

6.5. The CANTELLI Modification of the BOREL Zero-One Law . . . . . . . . . . . . . 154

7. BOREL'S earlier works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

7.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

7.2. BOREL (1903): F or es ha do wi ng s o f CANTELLI-like Re aso ni ng . . . . . . . . . . . . 157

7.3. BOREL (1905): An Early Identi fication of Geomet ric Probab ility with

Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

8. The BOREL-BERNSTEIN Polemic . . . . . . . . . . . . . . . . . . . . . . . . . . 1648.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

8.2. The Criticism: The Contribution of F. BERNSTEIN (1912) . . . . . . . . . . . . . 165

8.3. BOREL'S Response: BOREL (1912) . . . . . . . . . . . . . . . . . . . . . . . 171

8.4. Early Observations of LEBESOUE and LI;vY .. .. .. .. .. .. .. .. .. .. .. .. .. 175

9. Early Re-working of BOREL'S Strong Law . . . . . . . . . . . . . . . . . . . . . 176

9.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

9.2. The Contr ibuti on of G. FABER: His Query on the Relation of Probability to

Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

9.3. RADEMACHER and HAUSDORFF: The Evidence for the Evo lut ion of a Point of

View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

10. HAUSDORFF'S"Grundztige der Mengen lehre': A Notable Advance in Technique .. .. 182

10.1. General Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18210.2. FIAuSDORFF'S Proo f of the St rong Law: The Use of Mome nt s a nd BIENAYMt~-

TCHEBYCHEFF type Inequalities . . . . . . . . . . . . . . . . . . . . . . . . 183

10.3. [~AUSDORFF'S Method for Calculating Moments . . . . . . . . . . . . . . . . 186

10.4. HAUSDORFF'SContinued Fraction Theorem . . . . . . . . . . . . . . . . . . 188

Axiomatic Probability 125

O. Intro duction

Th e overa l l purpose of th i s work is to ske tch the h ighpoin t s in the dev e lop me nt

o f t he a x i oma t i c f o r m u l a t i on o f P r oba b i l i ty T he o r y i n t e r ms o f me a s u r e t he o r y . (I ti s no t our in ten t ion to deprec ia te the reby o the r e f for t s to found the theory of

p r oba b i l i t y on a x i oma t i c ba s es t ha t do no t e mp l oy t he c on c e p t o f me a s u r e t he o r y. )

This axio ma t ic for m ula t io n was f i rs t def ini t ively achie ved by A. KOLMOGOROV in

1933 . T he ne e d f o r a n a x i oma t i c f ounda t i on f o r P r oba b i l i t y T he o r y ha d be e n

s t ressed by HILBERT as par t of the s ixth pr ob lem in his cele brate d l i st of 1900: " . . . :

To t rea t . .. b y means of ax ioms , those phys ica l sc iences (sic) i n w h ic h ma t he m a t i c s

p lays an impor tan t pa r t ; in the f i r s t rank a re the theor ies of P robabi l i ty and

M ech an ics" (HILBERT (1900: 81)) . Thu s this w ork can b e con s ide red a sketch of the

hi s tory of pa r t o f one of HILBERT's problems . Her e we t race the h i s to ry f rom the

wo rk of BOREL (1909) thr ou gh the w ork of HAUSDORFF (1914); subse que nt ly

Par t I I wi ll ca r ry the na r ra t ive forw ard to STEINHAUS, FRI~CHET, CANTELLI,

POLYA, MAZURKIEWICZ and others , culm inat in g in the wo rk of KOLMOGOROV.

The l and ma rk paper , in i t i a ting the m od ern theo ry of probabi l i ty , i s E . BOREL's

" L e s P r oba b il i ti 6 s D 6n omb r a b l e s e t L e u r s A pp l ic a t i ons A r i t hm6 t i que s " o f 1909 .

H e r e w e a r e ma i n l y c onc e r ne d w i t h t he c on t e n t s , ba c kg r ound , a nd i mme d i a t e

reac t ions to th is paper . The key f igures imm edia te ly fo l lowing BOREL are

a . FABER, F. BERNSTEIN,and F. HAUSDORFF. Th eir c on tr ib ut io ns wi l l be dis -

cussed in turn. Key predecessors are A.WIMAN (1900, 1901) , E.VANVLECK

(1908) and, in fact , BOREL himself by vi r tue of a br ie f no te of 1905. Bo th WIMAN

and VAN VLECK are subjects of separa te b r ief note s ; the no te o n VAN VLECK has

alr ead y ap pe are d (NOVIKOFF & BARONE (1977)). O f these earl ie r works, o nly

BOREC's paper of 1905 is discussed a t any length here .

An y a deq ua te d i scussion of the conten t s of BOREL'S l and m ark pap er (which we

shal l refer to as BOREL (1909)) i s of necess i ty del icate an d som ew hat deta i led. T he

reason for th i s is the i ron ica l c i rcum s tance tha t BOREL, the un que s t ioned foun der o f

me a s u r e t he o r y , a t t e mp t e d i n 1909 t o f ound a ne w t he o r y o f " de n um e r a b l e

p r o b a b i l i t y " without r e ly i ng on me a s u r e t he o r y . T he i r ony i s f u r t he r c omp oun de d

in the l igh t of BOREL's paper of 1905 which ident i fi ed "con t inu ou s p rob abi l i t y" in

the uni t in te rva l wi th measure theory the re . Consequent ly , we a re a t g rea t pa ins

bot h to es tab li sh and to co mp reh end BOREL'S re luc tance in 1909 to accept the

unde r ly ing ro le of counta b le addi t iv i ty in h i s new theory . The f ir s t pa r t o f th i s paper

is thus a n a t tem pt to e xam ine BOREL's s ta te of mi nd in 1909, taking into ac co un t his

ea r l i e r ins ight s and h i s re luc tance to explo i t them.

BOREL's paper , "Probab i l i t6s D6 nom brab les" , fal ls in to th ree m ajor d iv i s ions :

a "genera l t he ory " (cu lmina t ing in wha t we have ca l l ed the BOREL Ze ro- On e Law) ,

an appl i ca t ion of th i s theo ry to dec imal an d d yadic expan s ions ( the BOREL S t rong

L a w , o r S t r ong L a w o f L a r ge N um be r s ) , a nd a s e c ond a pp l i c a t i on o f t h is t he o r y ,

t h i s t i me t o C on t i nue d F r a c t i ons ( t he B O R E L C on t i nue d F r a c t i on T he o r e m) .

Since BOREL'S pa pe r is as inte restin g for i ts defects as for its results (as Pr ofe sso r

M. KAC once rem arked , " a l l o f i ts theorem s a re t rue bu t a lm os t a l l o f the proofs a re

fa lse") , we summarize i t s shortcomings in Sect ion 4.6.

BOREL'S immediate successors devoted themselves to c lar i fying both BOREL'S

S t r ong L a w a nd h i s C on t i nue d F r a c t i on T he o r e m by t r e a t i ng t he m bo t h w i t h i n

12 6 J. BARONE ~f¢ A. NOVIKOFF

LEBESGUE'S the or y o f me asur e on the real l ine . 1 This c lar i f icat ion, howev er , was a t

the expense o f dras t i ca l ly d imini sh ing BOREL'S emph as i s o n the cons t ruc t io n o f a

ne w ge ne ra l t he o ry o f p roba b i l i t y c onc e rn i ng repeated independent trials.

STEINHAUS (1923) finally in co rp or at ed BOREL'S " g e n e r a l t he o ry" i n to me a s u ret he o ry by e xp l o i t i ng an a x i oma t i c c h a r a c t e r i z a t i on o f me a s u re t he o ry due t o

SIERPII~SKI (1918). STEINHAUS' w or k a nd m or e ge ner ally the in crea sing abs trac -

t ion of m eas ure the or y i t se l f ( ini t ia ted by FRt~CHET (1915) an d CARATHI~ODORY

(1914)) were events which helped pave the way to KOLMOGOROV's culminat ing

achievement . These deve lopments wi l l be d i scussed , among o the rs , in Par t

Whi l e w e ha ve r e s t r a i ne d ou r s e l ve s f rom d ra w i ng ge ne ra l h i s t o r i og ra ph i c

conc lus ions , we be l i eve the conten t s of ou r pap er (both he re and in Par t I I ) furn i sh

am m un i t io n for those w ho wish to i l lus t ra te ( i) DIEUDONNI~ 'S " fu s ion " hy pothes i s

of mathematical progress (DIEUDONNI~ (1975:537)), (i i) LAKATOS' thesis on the

gradu a l r igo r iz ing of pa r t i a l ly un der s to od reason ing v ia d i spute (LAKATOS (1963-

64) ) , and ( i i i ) the doc t r ine tha t " spec ia l problems c rea te genera l theor ies" . The

present s tudy of BOREL'S wo rk br ings to l igh t a s tupen dou s ins tance o f tunne l

vis ion. I t a lso shows that a foundat ional ques t ion (such as HILBERT'S) may have

unex pec te dly comp lex responses. Th e so lu t ion w e t race to HILBERT'S proble m did

no t c om e f rom a d i r e c t a t t a c k ; i t e vo l ve d f rom c om pu t a t i ons w h i c h s uc ce s sfu ll y

deal t w i th a ser ies of par t ic ular , wel l -c hose n ques t ions , f i rs t ra ised by BOREL.

1. Brief Sketch of Major Results of Borel (1909)

BOREL cons id e red an inf in it e sequence of tr i al s , each hav ing only tw o

poss ib le outcomes a rb i t ra r i ly ca l l ed ' " success" and " fa i lure" . The probabi l i t i e s of

"succ ess" and " fa i lu re" on the n h t r i a l a re Pn and qn, respec tive ly , where

0_-<pn, qn<l,

p , + G = l , n = 1 ,2 , 3 , .. . .

Th e tr i a ls a re as sumed indep enden t . In con tem po ra r y t e rms such t ri a ls a re ca l led

"Binomia l " or "PoISSON" t r i a l s . In wha t represent s a dec i s ive new s tep , BOREL

asked for the p rob abi l i ty A k tha t exac t ly k successes occu r in such an inf in i t e

sequen ce (k = 0 , 1, 2 , . .. ) and , mo s t im por ta n t , the p roba bi l i ty A , of the occ ur ren ce

of inf in i t ely m an y successes.

There i s no nota t ion for the se t s (or "event s" ) under cons ide ra t ion in BOREL

(1909). In consequ ence , the re a re a l so no u nions , com plemen ts , in te rsec tions , etc.,

ind ica ted or re fe r red to as such .

1 T h i s s t a t e m e n t m u s t be qua l i f i e d by e m p ha s i z in g f ir s t t ha t N . WIE NE R a nd P . L ~ vY w or ke d a t a

different level, not follow ing BOREL'S lead directly, an d sec ond POLYA, CANTELLI an d MAZURKIEWlCZ

c ons ide r e d r a the r ge ne r a l " r a nd om va r i a b l e s " o r t r i a l s " a s i n B OR EL 'S "ge ne r a l t he o r y" , w i tho u t

a ppe a l ing t o m e a su r e t he o r y .

Let us in t roduce the nota t ion , the re fore , tha t

E k= se t o f a l l sequen ces wi th exact ly k successes,

k = 0 , 1 ,2 . . . . .

E~ = se t of al l s equences wi th inf in i te ly many successes .

BOREL then sought express ions for

Ak = P(Ek) , k = O, 1, 2 ,. . . ,

A~ =P(E~)

in t e rms of the g iven num er ica l s equence {p,} , n= 1 ,2 , . . . . BOREL pers i s t en t lyoo

t rea ted separa te ly the cases in which ~Pn converged or d ive rged•1

Th e f i rs t resul t, c onc ern ing th e case k = 0, asser ted tha t

A o = (1 - p~ ) (1 -P 2 ) . . . (1 - p , , ) . .. .

Here the r ight -hand s ide represent s a convergent non-vani sh ing inf in i t e produc t

(f ~ p n c onve rge s a nd i s t o be i n t e rp r e t e d a s z e ro if ~ p , d i verge s a nd he nc e1 1

A I = ~ A o Pi _ Ao ul1 1 - P i 1

1 - p i

The se ri es on the r ight i s convergen t i f ~P n converges , whi le i t is to be

inte rpr eted as zero i f ~ Pn diverges .1

More general ly, for any f ini te k, BOREL asser ted that

A k = ~ A o u i u i 2. ., ui~, t h e s u m m a t i o n o v e r l < i l < i 2 < . . . < i k < o o .

He re the r igh t-ha nd s ide is to be inte rpr eted as zero i f ~ Pn diverges , bu t BOREL

asse r ted tha t in the conv ergen t case 1

Final ly, le t t ing

0 < A k < l , k = 0 , 1 ,2 ,3 . . .. .

S = A o T A 1 + . . . + A k + . . . ,

128 J. BARONE& A. NOVIKOFF

a3 o:3

BOREL sh ow ed tha t i f ~ p , co nve rge s , the n S = 1, whi le i f ~ p , diverges , S = 0. S ince1 1

A~o = 1 - S , t h e r e m a r k a b l e r e s u l t ( " B O R EL ' s Z e r o - O n e L a w " )

/°lf ~ p n c o n v e rg e s

A o 3 ~ 1oo

if ~ pn dive rges .1

is o b t a in e d . T h e r e st o f t h e p a p e r c o n t a i n s , a s i ts i m p o r t a n t r e su l ts , t w o a p p l i c a t i o n s

o f t h is c u r i o u s b e h a v i o r o f A ~ .

T h e f i rs t a p p l i c a t i o n c o n c e r n s t h e d y a d i c e x p a n s i o n o f a r e a l n u m b e r x c h o s e n

" a t r a n d o m " i n [ 0 , 1 ] :

x = . b l b 2 . . . b , . . . - ~ - b n1 2n

w h e r e b , = b,(x) i s e i t he r 0 o r 1 . I t is a s s um ed tha t t he se quen ce {b ,} is genera t ed , o r

t h e n u m b e r x is " c h o s e n " , s o t h a t e a c h b i n a r y d i g i t b , (x) h a s p r o b a b i l i t y ½ o f b e i n g 0

o r 1, a n d a l s o s o t h a t t h e v a r i o u s d i g it s n = 1, 2 , 3 . . . . a r e e x a m p l e s o f i n d e p e n d e n t

t r ia l s . BOREL cho ose s a f i xed sequen ce 2 , go ing t o i n f in i t y wi th n bu t t h a t

£ = 0i m ] / n "

L e t v, (x) d e n o t e t h e n u m b e r o f o n e s a m o n g t h e f ir st n b i n a r y d i g it s b~(x),b z (x ) , . . . , b n (x ) . F o r e a c h d y a d i c e x p a n s i o n B O R EL c o n s i d e r e d t h e a s s o c i a t e d

sequ ence o f " t r i a l s " t he n th t r ia l o f wh ich i s de f ined as success i f an d on ly i f

Ivz,(X)-nl~£~, n=1,2,3,....

L e t Pn b e t h e p r o b a b i l i t y o f t hi s o c c u r r e n c e a n d q , = 1 - p , t h e c o m p l e m e n t a r y

p r o b a b i l i t y . ( W e h a v e r e v e r s e d B O R E L 's n o t a t i o n f o r p , a n d q n t o c o n f o r m w i t h o u r

n o t a t i o n f o r Eoo.) B y a n a p p l i c a t i o n o f t h e C e n t r a l L i m i t T h e o r e m , B O RE L a t t e m p t sco

t o e s t ab l i s h t h a t ~ p , c o n v e r g e s , a n d h e n c e b y a n a p p l i c a t i o n o f h i s m a i n e a r l ie r1

r es u lt , t h a t Aoo = P ( E ~ ) = 0 . H e c o n c l u d e d t h e n t h a t t h e e ve n t,

h a s p r o b a b i l i t y 1 , i.e.,

lim Vzn(X) _ 1

,~oo 2n 2'

1 The formulas for A0, A 1,..., Ak, ... are extensions o f formulas for finite numbers of trials to thecase of denumerably many. The case of A~ is utterly new, since E~ is empty for a finite number of trialsand in that case its probability is of no interest.

This resu l t is ca lled var ious ly BOREL's Law of Large Num bers , the S t rong L aw of

Large Num bers , or BOREL's Law of No rm al Num bers . BOREL'S a rgu me nt wi ll be

anal yzed in § 5.

The second appl ica t ion cons iders the expan s ion in cont in ued f rac t ions of a" r a n do m l y" c hose n i r r a t iona l num be r c hose n f r om [ 0, 1 ]:

a l + l

a 2 + 1

a 3 + 1

a . +i

° , .

Here each a , = a,(x) is a pos itive integ er, n = 1, 2, 3 .. ..

BOREL cons tructe d a sequence of tr ia ls , and associa ted "s ucce ss" an d

"fai lu re" , by ch oosin g a f ixed sequence ~b(n) and def ining for each ir ra t ion al x the n h

" t r ia l" as be ing a "success" or " fa i lure " (wi th cor responding probabi l i tie s p , , q , )

according as a ,>qb(n) or a , <¢(n) . Thus BOREL has assoc ia ted wi th each inf in ite

cont in ued f rac t ion an ins tance of an inf in i te sequence of tr ials , each t r ia l having

only two outcomes; he then seeks to apply h is Zero-One Law ment ioned above

to this collect ion of sequences of t r ia ls .By adro it m ani pu lat io ns of con tinu ed fract ions (see below), BOREL established

the inequali t ies

2 1 33 ( k + l ~ < P(a .(x)> k) < k +~"

Repla cing k by q~(n) in this ineq uali ty sh ows tha t ~ p. converges if and only if

~ ~ j - does . 1Ag ain ap pealing to his earl ier result concerning A ~, BOREL asser ted1

1if ~ ( n ) converges, then with pro bab il i ty 1, a , wil l ul t im ately sa t isfy a , < qS(n)

whi le

1if ,~O~n) diverges, the n with prob abil i t y 1, a , wil l inf ini te ly of ten vio la te

tha t inequa l i ty .

This result w il l be cal led hencefo r th BOREL'S Continued Fraction Theorem.

2. "Countable Independence" as a Key Principle

As is evident f rom this br ief sketch, the proper ty of independence of tr ia ls

under l ies the formulas for Ao, A1, . . . ,A k, . . . and A~, and hence a lso the two

a pp l i c a t i ons . We p r opos e t o u s e t he ph r a s e "countable independence" for the

pr inc ip le tha t BOREL expl ic i t ly in t rod uce d a nd on which a l l o f h i s resu l t s a re based .

Thi s i s the as se r t ion , usua l ly t ak en as a hypothes i s , tha t a g iven co l l ec t ion of events ,

B1, B2 . . . . . Bn, . . . sa t is fy

P ( ~ B i ) = O P (B I). (2.1)

W hen the co l l ec t ion of event s is fin it e , the cor re spon ding pr inc ip le was know n as

the " lo i des probabi l i t6s com pos6es" , a l th oug h the no ta t io n for s e t in te rsec t ion was

not genera l ly emp loyed . BOREL assumed the pr inc ip le i f the event s B~ re fe r red to

d i f fe ren t t r ia l s for d i f fe ren t i and , m os t im por ta n t , a s sumed i t to h o ld even i f the

range of the index were inf in i te . A par t i cu la r ins tance o f spec ial impo r tance , which

we might ca ll the limited pr inc ip le o f countab le ind ependen ce , i s tha t (2 .1) ho lds i feach P(BI)= 1.T he r e a de r c a n s e e a n a na l ogy ( w h ic h w e be l ie ve mus t ha ve a c t e d pow e r f u l ly on

BOREL) wi th the b eh av ior o f length app l ied to d is joint intervals , B~, on the real l ine 1

This ex tens ion to the inf in i t e range and espec ia lly the in te rp re ta t io n o f the l e f t -handoo

s ide as a sor t o f genera l i zed l ength i f @ B i was no t i t se l f an in te rva l (or even

1express ible as a f ini te un ion of intervals ) l ies a t the hea rt of BOREL'S ear l ier ,

p r o f o und l y i mpo r t a n t d i s c ove r y o f t he t he o r y o f me a s u r e . I n pa r ti c u l ar , t he t he o r y

of measure , by focuss ing on th i s pr inc ip le ( "cou ntab le addi t iv i ty" ) i s l ed to e x tend

the c lass J o f in te rva l s to a mu ch wider and mo re s ign i f ican t cl as s N , wi th the key

pr op er ty tha t i f each of the Bi i s a se t of this w ider c lass N, and the B~ are d is joint ,oo

@ Bi is necessar i ly wi thin the c lass N.1

BOREL in 1909 may have fe lt h imse l f em bark ing on a s imi la r explora t ion , us ing

indep end ence o f t r ia l s a s the co unt e rp ar t to d i s jo in tness of in te rva l s and wi th

num e r i c a l p r o duc t s r e p l a c ing num e r i c a l s ums . I nde e d , t he ve r y na me B OR EL gavehi s theory , denumerable probab i l i ty , re fe rs prec i se ly to the ran ge of the index i in the

" lo i des probabi l i t~s com pos6es" . I t mos t ce r t a in ly does not r e fe r t o t he num be r o f

poss ib le d i s t ingui shable ou tcomes (i.e. sequen ces of t r ia ls ) as BOREL him self wel l

knew. (Fo r examp le , BOREL explo i t s the dy adic exp ans ion of num bers in the uni t

in te rva l a s an exam ple of h i s theory . The co l l ec t ion of such expan s ions has the

c a r d i na l num be r o f t he c on t i nuum. ) S om e how B OR EL f el t t ha t " de num e r a b l e

prob abi l i ty" , h is new theory , was poi sed be tween c las sica l " f in i t e pro bab i l i ty" an d

" ge o me t r i c p r oba b i l i t y " . G e om e t r i c p r oba b i l i ty , he w e ll kne w (cf BOREL (1905)),

ma y be ge ne r a li z e d to m a ke u s e o f h is ow n t he o r y o f me a s u r e a nd e ve n to m a ke u s e

o f L E B E S G U E ' S ne w i n t e g r a t i on t he o r y ( na me l y f o r t he c om pu t a t i o n o f me a nva lues ). In add i t ion to a " lo i des probab i l i t6s com pos6es" , c l as sica l p rob abi l i ty a l so

1 W h e n a u n i o n of disjoint se t s i s t a k e n , t h e sy m b o l @ A k wil l be used in s tead of U Ak t o e m p h a s i z e

th a t t h e A k are (pair-wise) disjoint .

has a " lo i des probabi l i t6s to ta les " :

the e vents B~ being assum ed dis joint (i.e. mut ua l l y i nc ompa t i b l e ) . I n s ome ke y

ins tances BOREL exte nd ed this a lso to th e inf ini te range,

P @ Bi) = ~ P(Bi) (countable addi t ivi ty) ,

but , as we shal l argue, a l l the internal evidence is that BOREL regarded countableindependenceas the essent ia l , new (and proba bi l is t ic) ingre dient o f his new theory .

By cont ras t he used countableadditivityse ldom (of ten sur rep t it ious ly) , and he never

explored i t s impl ica t ions ; about i t he had rese rva t ions so deep tha t he f requent ly

of fe red "a l t e rna t ive" proofs to evade re l i ance on i t .

This assessment , w hich we shal l defend by sui table analys is , explains a t leas t in

pa rt wh y BOREL fai led to dra w the co nclus ion , a t t r ib ute d to CANTELLI (1917a,

1917b), thatco

A c o = 0 i f ~ p , c o n v er g es1

even i f the t r ia ls are not a s s ume d i nde pe nde n t .

BOREL's fasc ina t ion wi th the pr inc ip le of counta b le indepe nden ce s imi lar ly

may expla in h i s fa i lu re to use anywhere n BOREL (1909) an arg um en t that

P(~Bi)<~P(Bi) (countab le sub-addi t iv i ty)

which fo l lows f rom coun tab le addi t iv i ty even i f the B~ a re not mut ua l l y i nc om-

pa t ib le . Wh ene ver an event is shown to hav e pro babi l i ty 0 , i t is no t b y pro of tha t i t

ha s " s ma l l c ove r s ", bu t r a t he r by p r oo f t ha t i t s c om pl e m e n t ha s p r ob a b i l i t y 1 . T h i s

l a t te r r e f o r m u l a t i on be c om e s a n a s s e rt i on a bo u t t he p r oba b i l i t y o f a n i n t e r se c t i on

and rest s on the pre fe r red pr inc ip le of coun tab le indep endence , o f t en in the

" l i m i t e d" f o r m.W e shal l sho w below (§ 7.2) tha t BOREL kn ew the CANTELLI pa rt of the BOREL-

CANTELLI lem ma s as ear ly as 1903, but in the co nte xt of a geom etr ic , not an

abs t rac t , space . M ore exac t ly , BOREL assoc ia ted th i s type of reasoning wi th the

HEINE-BOREL Cove r ing T he ore m as a pre l iminary . Thi s fur the r suppor t s ou r the -

s is tha t BOREL did n ot s ee the fu ll ana logy of prob abi l i ty w i th measu re except when

t h e p r o b l e m p e r m i t t e d a geometric interpretation.In our op in ion , the ana logy was

imper fec t ly seen even the n (cf the discuss ion of his exch ang e wi th F. BERNSTEIN,

§ 8).

3. Denumerable Probability versus Measure Theory

Befo re we exa min e BOREL's text in some deta i l , i t i s ins t ruct iv e to su m ma rize a

few more recent ly acqui red ins ight s . In contempora ry t e rms , each " t r i a l " cons id-

ered by BOREL is a 2-p oin t m easure space .Th e inf in it e sequences of ou tcom es a re jus t

poin t s of the (den ume rable ) CARTESIAN pro du c t of such 2-p oin t spaces. To each

sequence o f ou tc om es can b e as soc ia ted a sequence o f 0 's and l ' s, where fa i lu re

co rre spo nd s to 0, say, and success to 1. This pro vide s a m ap pin g fr om /~ {0, 1}~,

whic h is the CARTESIAN pr od uc t o f 2-p oint spaces , to the uni t interv al . This

ma p p i ng is 1 - 1 e xc e p t t ha t e a c h dya d i c r a t i ona l nu mb e r ma y ha ve t w o d i s t i nct

pre - images , one t e rmina t ing in 0 ' s , the o the r in l ' s . Fur the r , the as sumed

independence be tween d ig i t s un ique ly de te rmines the probabi l i ty , o r a s we would

say equiva len t ly now, measure , o f each dya dic in te rva l [Pn , P -~- I ] ; t h i s measur e ~

can be shown to sa t i s fy

m ( ~ B i )= ~ m(Bi)1

i f each B i is a d yad ic interv al , a nd @ B i i s again a dya dic in terval . F inal l y i t fol lows1

fro m al l this (by ins ights gaine d af te r 1909) that th e sequ ence {Pn} dete rm ines a a-

addi t ive measu re on the a -a lg ebra o f "BOREL se t s " N of [0 , 1 ]. I f p = l , n

= 1, 2 , 3 , . . . , t hen the m easure i s the ve ry one in t ro duc ed ea r l i e r by BOREL himsel f.

I f the {p,} a re no t iden t i ca l ly ½, the co r resp ond ing measu re i s a va r i an t o f the ab ove

(of the typ e la ter envis ioned by RADON (1913)) , a sor t of STIELTJES me asu re

assoc ia ted wi th a su i t ab le d i s t r ibu t ion func t ion F(x) bu t s t i ll def ined a t leas t on theBOREL se ts of [0, 1 ] . Thu s BOREL'S new th eor y o f" de nu m er ab le pro bab i l i ty" is, i f

each p , =½, es sent ia l ly a d i sgui sed ve rs ion of h i s own ea r l i e r theory of mea sure in

the uni t in te rva l , the proba bi l i s t i c im pac t of which h e rea l i zed a t l east pa r t i a l ly

w h e n t h e p r o b l e m w as o n e o f " g e o m e t r i c p r o b a b i l i t y " (cf § 7.3). There is no

evidence in favor of the sup pos i t ion tha t BOREL un der s too d th a t the case of genera l

{p ,} a l so gave r i s e to a "genera l i zed" BOREL measure (i.e., a d i f fe ren t countab ly

addi t ive m easure on the same cou ntab ly add i t ive f ie ld of sets). Th ere i s ev idence

t ha t i n t he c a s e p , =½ , n=1 , 2 , 3 , . . . , h e d i d s e ns e t he c onne c t i on be t w e e n

" p r oba b i l i t 6 s d dnom br a b l e s " a nd h is ow n t he o r y o f me a su r e . I n t hi s c a se he s pea ks

of the "p oin t de vue g6om 6t r ique" , re fe r r ing to the in te rva l [0 , 1 ] wi th i t s a s soc ia tedBOREL measure , and the "p oin t de vu e log ique" , re fe r r ing to the inf in i t e s equences

of t r ia ls , which w e deno te X {0, 1}i. BOREL asser ts that on e can em plo y e ither . (Cfi=1

§ 5.1 for the exact c i ta t ion. ) He m ake s this co m m en t o nly in discuss ing the specif ic

c a se p , =½ , n = l , 2 , 3 , . . . . O t he r i n t e r na l e v i de nc e t o s up por t a ll o f t he a bov e

asse r t ions concern ing wha t BOREL rea l i zed or rea l i zed imper fec t ly , wi l l be

di scussed be low.

Th e c ruc ia l ev idence aga ins t a s se r ting tha t BOREL rea l i zed tha t " de nu m era ble

p r o b a b i l i t y " was mea sure t heo ry in [0 , 1 ] is h is repea ted evas ion of counta b le

addi t iv i ty and cou ntab le sub -addi t iv i ty in a lm os t a l l o f h i s reasoning . Thi s ev idence

seems to us decis ive . A lesser evide nce of the same sort i s his seeing som ethi ng n ew

in the fac t tha t s e t s o f pro bab i l i ty ze ro ne ed no t be em pty . An d of course the re i s h i s

own as se r t ion ( in the face of h i s own con t ra ry as se r t ion for the case p ,=½, n

Axiom atic Probability 133

= 1 , 2 , 3 . . . ) tha t "d en um era b le p ro ba bi l i ty" i s in genera l a theo ry intermediate to

f in i te o r c o m b i n a t o r i c p r o b a b i l i t y , a n d g e o m e t r i c o r c o n t i n u o u s p r o b a b i l i t y . 1

We turn no w to BOREL'S Cha pte r I (pp . 247-257) to jus t i f y the abo ve

i m p u t a t i o n s .

4. The Evidence From Borel's Chapter I

4.1. Denumerable Probability Contrasted with Continuous Probability

" On e g e n e r a l l y d i s t i n g u i s h e s , i n p r o b a b i l i t y p r o b l e m s , t wo p r i n c i p a l

ca tegor ies , accord in g to w he the r the n um be r o f poss ib le cases i s f ini t e o r

in f in i te : the f i rs t ca teg ory co ns t i tu tes wh a t o ne ca l l s discontinuous p robabilities,

o r p r o b a b i l i t i e s i n a d i s c o n t i n u o u s d o m a i n , wh i l e t h e s e c o n d c a t e g o r y

c o m p r i s e s continuous probabilities o r geometric probabilities. Such a classif i-

c a t i o n a p p e a r s i n c o m p l e t e wh e n o n e r e f er s b a c k t o t h e r e su l ts a c q u i r e d i n th e

theo ry of s e t s ; be tw een the ca rd in a l i ty o f f in i te s et s and the ca rd ina l i ty o f the

c o n t i n u u m s t a n d s th e c a r d i n a l it y o f d e n u m e r a b l e s e ts ; I p r o p o s e t o s h o w b r ie f ly

t h e i n t e re s t wh i c h i s a t t a c h e d t o q u e s t i o n s o f p r o b a b i l i t y i n wh o s e s t a t e m e n t

such se ts intervene; I wil l ca l l them, for shor t , denumerable probabilities.

Before de f in ing more prec i se ly denumerab le p robabi l i ty , I wish to ind ica te

in a f ew word s the r easons aga ins t f u r the r f a il ing to s tudy i t. P r inc ipa l am on g

t h e se r e a s o n s i s th e i m p o r t a n c e o f t h e n o t i o n o f d e n u m e r a b l e s e ts ; th i s

i m p o r t a n c e wa s n o t c o n t e s t e d b y a n y m a t h e m a t i c i a n ; b u t i t s e e m s t o m e t o b e

grea te r s t i l l than one be l ieves .

M an y an a lys t s , indeed , pu t in the fi r st r an k the idea of the con t inu um ; i t i s

th i s concept which in te rvenes more or l e s s exp l ic i t ly in the i r r easoning . I have

i n d i c a t e d r e c e n t l y h o w t h is n o t i o n o f t h e c o n t i n u u m , c o n s i d e r e d a s h a v i n g a

c a r d i n a l i t y g r e a t e r t h a n t h a t o f t h e d e n u m e r a b l e , s e e m s t o m e t o b e a p u r e ly

n e g a t i v e n o t io n . T h e c a r d i n a l i t y o f d e n u m e r a b l e s et s a l o n e b e i n g wh a t we m a y

k n o w i n a p o s i t i v e m a n n e r , t h e l a t t e r a l o n e i n t e r v e n e s effectively in our

r easonings . I t is de a r , indeed , tha t the se t o f ana ly t ic e leme nts tha t can be

a c t u a l l y d e f i ne d a n d c o n s i d e r e d c a n b e o n l y a d e n u m e r a b l e s e t; I b e l ie v e t h a t

t h is p o i n t o f v i e w wi ll p r e v a i l m o r e a n d m o r e e v e r y d a y a m o n g m a t h e m a t i c i a n s

a n d t h a t t h e c o n t i n u u m wi ll p r o v e to h a v e b e e n a t r a n s i t o r y i n s t r u m e n t , wh o s e

prese n t -da y u t i l i ty i s no t neg l ig ib le (we wil l supp ly ex amp les a t once) , bu t i t wi l l

c o m e t o b e r e g a r d e d o n l y a s a m e a n s o f s t u d y i n g d e n u m e r a b l e s et s, wh ic h

cons t i tu te the so le r ea l i ty tha t we a r e cap able o f a t t a in ing . " BOREL (1909: 147-

The se ope ning word s ind ica te tha t BOREL be l ieves tha t the se t o f poss ib le

outcomes which he wi l l d i scuss and which in modern te rms i s h i s s ample space , i s

d e n u m e r a b l e . No t h i n g c o u l d b e m o r e m i s l e a d i n g : t h e s a m p l e s p a c e s h e d i s c u s s e s

a r e a l wa y s d e n u m e r a b l y i n fi n it e p r o d u c t s o f f in it e, o r a t m o s t d e n u m e r a b l y i nf in i te ,

f ac tor spaces. I ndeed , ev en the s imples t o f these, the den um erab leCARTESIAN

1 Ind eed the class of denumerable sam ple spaces is intermediate between finite and non -

denumerable ones. But the space /~ {0, 1}i under conside ration in 1909 is not among them, being ratheri = 1

a different way of viewing the un it interval [0, 1].

p r o d u c t o f 2 - p o i n t s p a ce s , is n o n - d e n u m e r a b l e , a s h a d b e e n s h o w n e a r l i er b y

G. C ANT OR . T h e r e p r e s e n t a t i o n o f t h e u n i t i n t e r v a l a s a p r o d u c t o f 2 - p o i n t s p a c e s is

a b r i l l i an t idea , bu t i t cann ot a l t e r i ts fami l ia r , and to BOP,EL r epug nant , ca rd ina l

n u m b e r c . T h e e v i d e n c e f o r t h e r e p u g n a n c e , a n d t h e m i s c o n c e p t i o n a s t o t h ec a r d i n a l n u m b e r o f t h e s a m p l e s p a c e s c o n s i d e r e d r e - e c h o e s i n th e c l o s i n g li n es o f

t h e p a p e r :

" A t s u c h t i m e a s t h e t h e o r y o f d e n u m e r a b l e p r o b a b i l i t i e s is d e v e l o p e d i n t h e

m a n n e r j u s t i n d i c a t e d , i t w il l b e i n t e r e st i n g t o c o m p a r e t h e r e s ul t s s o a c q u i r e d

wi t h t h o s e o b t a i n e d i n t h e t h e o r y o f c o n t i n u o u s o r g e o m e t r i c p r o b a b i l i t y .

I n t h e g e o m e t r i c c o n t i n u u m t h e r e exist cer ta inly ( if i t i s no t a misu se to

e m p l o y t h e v e r b to exist) s o m e e l e m e n t s wh i c h c a n n o t b e d e f i n e d : s u c h i s t h e

r e al s en s e o f t h e i m p o r t a n t a n d c e l e b r at e d p r o p o s i t i o n o f M r . G e o r g C a n t o r : t h e

c o n t i n u u m is n o t d e n u m e r a b l e . S h o u l d a d a y c o m e w h e n t h e s e undefinablee leme nts cou ld be pu t a s ide as no longe r neede d m ore or le s s impl ic i t ly , it wo uld

c e r t a i n ly b r i n g g r e a t s i m p l i f i c a t i o n in t h e m e t h o d s o f An a l y s i s ; I s h o u l d b e

h a p p y i f t h e p r e c e d i n g p a g e s c o u l d h e l p a r o u s e t h e i n t e r e st wh i c h t h e s t u d y o f

such ques t ions deserves ." ( I ta l ics in the or iginal . ) BOP,EL (1909: 271).

On e o f the b es t wa ys to fa i l to see a re la t io n (e.g . tha t X {0, 1}~ an d [0, 1] arei=~

equiva len t a s ca rd ina l s e t s and even as measure spaces ) i s to yea rn tha t no such

r e l a t i o n h o l d. B ORE L s u r e ly d i d n o t w i s h h i s o wn " d e n u m e r a b l e p r o b a b i l i t i e s " t o

jo in the con t inuum as a fu ture fos s i l , a mere t r ans i to ry dev ice .

4.2. The Calculation o f A o

BOREL turned f i r s t to e s tab l i sh ing the fo rmula fo r Ao

A 0 = (1 - p~)(1 - P2) -.. (1 - p, ) .. .. (4.1)

BOREL exc ludes in advan ce a ny p , = 1 so he can conc lude , i f~ . p , is conve rgent , tha t1

0 < A 0 < 1. ° ' I n the case of convergen ce , the ex tens ion of the p r inc ip le o f com po s i teprob abi l i t i e s goes wi tho ut s a y ing . . . " (BOREL (1909: 249). Fur t he r s ince the l imi t o f

the pa r t i a l p rod uc ts i s pos i t ive , they ap pr oa ch the i r l imi t wi th smal l r e la t ive e r ror a s

wel l a s abso l u te e r ror . I n conc lus ion , " . . . ; the pass age to the l imi t tha t w e have

pe r for me d thus does no t r a i se any d i f fi cu lt i es an d i s en t i r e ly jus t i fi ed . " BOREL

(1909: 249).

In f ac t , wha t BOREL i s sk imming over i s the l imi t r e la t ion

2 i m P ( @ B i ) = P ( @ B i ) . (4.2)

An a s s u m e d i n d e p e n d e n c e a s s u re s a d d i t i o n a l l y t h a t i f e a c h B i h a s p r o b a b i l i t y

qi, t h e nn

P(~11Bi)=I~ P(B~)=~[(1-p~)'I

Axiom atic Probab ility 135

T h e l i m i t r e l a t i o n (4 .2 ), h o w e v e r , h a s n o t h i n g t o d o w i t h i n d e p e n d e n c e ; i t is o n e o f

t h e m a n y c o n s e q u e n c e s , a n d e v e n e q u i v a l e n t fo r m s , o f c o u n t a b l e a d d i ti v i ty . T h e

l i m i t r e l a t i o n (4 .2 ) is, f o r B O R E L, b o t h t o o d e s i r a b l e t o b e f a l s e a n d t o o e v i d e n t t o

r e q u i r e d i s cu s s io n o r e l a b o r a t i o n . I n d e e d , s in c e h e n o w h e r e e m p l o y s a n o t a t i o n f o rt h e a l g e b r a o f s e ts o r e v e n f o r s e ts t h e m s e l v e s o r f o r s e t f u n c t i o n s , it w o u l d n o t h a v e

b e e n e a s y f o r h i m t o s t a t e e x p l i c i t l y . H a d h e e m p l o y e d t h e s y m b o l i s m P ( @ B i ) ,

p e r h a p s h e m i g h t h a v e b e e n d r i v e n t o q u e s t i o n t h e d o m a i n o f t h e s e t f u n c t i o n P ( - )

j u s t a s h e h a d e a r li e r q u e s t i o n e d a n d e x t e n d e d t h e d o m a i n o f " l e n g t h " f o r p o i n t- s e t s

in [0, 1] .

T h e d i v e r g e n t c a s e h er e , a n d l a te r , c al ls f o r sp e c i al " p r e c a u t i o n s " . F o r

" p r o b a b i l i t 6 d i s c o n t i n u e " ( i . e . , f i n i t e s a m p l e s p a c e s ) p r o b a b i l i t y z e r o m e a n s

i m p o s s i b i l i ty . F o r " p r o b a b i l i t 6 c o n t i n u e " t h i s is n o t a b l y f a ls e , a n d B O R E L r e f e rs t h e

r e a d e r t o h i s o w n p a p e r o f 1 90 5. I n t h i s p a p e r ( c f § 7 .3 ) he ha d def ined t h e g e o m e t r i c

p r o b a b i l i t y o f B O R E L s e ts o n [ 0 , 1 ] a s b e i n g t h e i r m e a s u r e . T h i s e x t e n d s it s fa m i l i a r

d e f i n i t io n . I n t h is w a y h e p r o v e s , f o r i n s t a n c e , t h a t t h e p r o b a b i l i t y f o r a n u m b e r

p i c k e d a t r a n d o m t o b e i r ra t i o n a l is z e r o , e v e n t h o u g h t h is o u t c o m e is n o t

i m p o s s i b l e . H e n o w s a y s i t i s t h e s a m e i n " p r o b a b i l i t 6 d d n o m b r a b l e " : i n t h e

d i v e r g e n t c a s e t h e f o r m u l a ( 4 . 1 ) f o r A 0 g i v e s t h e v a l u e z e r o , s i n c e t h e r e l a t i o n

l i m 1 ~ ( 1 - p i ) = A on ~ o 3 1

is e v i d e n t l y a c c e p t e d , d e s p i t e t h e f a c t t h a t t h e s e q u e n c e o f u n b r o k e n s u c c e s se s

h a p p e n s t o e x is t. N o n - e m p t y s ets m a y h a v e z e r o - " p r o b a b i li t6 d 6 n o m b r a b l e " j u s t a s

n o n - e m p t y s e t s ( f o r e x a m p l e , t h e r a t i o n a l s ) m a y h a v e m e a s u r e z e r o ; h e d o e s n o t

c l a i m t h a t t h e y a re s e ts o f m e a s u r e z e r o s i n c e h e i n t r o d u c e s n o m e a s u r e .

4.3 . The Calculation o f A1 , A2, and A k

T h e d e r i v a t io n o f t h e f o r m u l a f o r A 1 i n v o lv e s a m o r e o v e r t u s e o f c o u n t a b l e

a d d i t i v i t y , b u t , l i k e t h e h i d d e n u s e i n t h e l i m i t r e l a t i o n ( 4 . 2 ) , t h i s a p p l i c a t i o n

p a s s e s w i t h o u t c o m m e n t . I f t h e l o n e s u c c e s s i n a s e q u e n c e o f t r ia l s i s a t t h e n h

t r ia l , t h e n t h e p r o b a b i l i t y o f t h i s s e q u e n c e i sG

A o = p , F I (1 - p i ) ,co~ 1 - P n i = 1

i ~ - n

i n t e r p r e t e d a s z e r o i n t h e d i v e r g e n t c a s e. I t fo l lo w s , b y t a c i t u s e o f c o u n t a b l e

a d d i t i v i t y , t h a t A a i s th e s u m o f th e s e c o,. I n t r o d u c i n g t h e n o t a t i o n

P ,U n - -

1 - p , 'w e h a v e

A1 = ~ (Ao u,) (4.3)1

which BOREL wr i t e s a s c~

A o ~ u, . (4.4)1

136 J. BARONE~: A. NOVIKOFF

This raises the q uest ion: in the dive rge nt case, is A ~ a sum of zeros as in (4.3) , and

hen ce zero , or i s it a pro duct of (0) (Go) as in (4 .4), and h ence indeterm inate? ~ in fact

BOREL does no t f ind the former reason ing , a rout ine ex amp le o f countab le

addi t iv i ty , compe l l ing :

. . . On the other hand, one can say that the des ired probabi l i ty Aa i s the sum of

separate p robabi l i t ies co, each of whi ch i s zero; how ever , s ince they are not f in ite

in number , one cannot conc lude wi thout spec ia l care tha t the to ta l probabi l i t y

i s zero , in v i ew o f the fact tha t zero probabi l i t y does no t denote imposs ib i l i t y .

BOREL (1909" 250).

This i s an interest ing lapse: there i s no reason why a sum o f zeros can fa i l t o be zero ,

unless i t be that

i.e. coun table addit iv i ty i s expl ic i t ly false," BOREL'S subsequent argument i s to

coun ter just that poss ibi l i ty . This i s ev ide nce that cou ntab le ad dit iv i ty was no t a

presup posed t ra it o f probabi l it y . The rea l i za t ion tha t a probabi l i t y o f zero need no t

mea n imposs ib i l i t y shou ld not l e ssen one ' s conf idence in countab le add i t iv i ty so far

as we can see .

BOREL, in any case , at tempts an a l ternat ive argument to show that in the

diverge nt case A 1 = 0 . Let o-, be the pr obab i l i ty that there i s exact ly one success in

the f irst n trials . Then BOREL asserts that

a . = (1 - P l ) - . . ( 1 - p n ) ( U l q - " " q - U n ) < e - ( m + ... + p n ) ( u I q _ . . . _}_Un

and hence tha t l im a , = 0 .

The conc lus ion , thou gh t rue , does n o t fo l l ow f rom the g iven upper e s t imate . 2

He conc ludes " In t he d ive rgen t case."

1 S i n c e y ~ u , = o o i f ~ G = o o .

2 T o sh ow that BOREL'S orig inal uppe r est ima te is defic ient in i ts intended pu rpose, co nsider, for

example , the choicee 2n 1 u =p,,= e2."

P , = ~ + e Z , , q , l + e 2 , , q,co

Th e n ~ p , i s divergent and n1 ~Uk~btn~e2n,

w h i l e n

~ p k < n ,

so that e -(p~ +"+P~)(u1 +u 2 + - . - +u , )> e - " e 2~= e' .

This sho ws that for at l east some choice o f p , such that ~ p, i s diverge nt e - ~ P ~ u need no t tend to zero .

1 1I n fac t th i s whene ver the su f f i ci ent ly rap id ly to 1 , which i s sure ly cons i s tent wi tha p p e n s Pn c o n v e r g e

the demand for d ivergence of ~ p , . I t i s true , howeve r , that l im cr = 0, as is establ ished in BARONE1 n~oa

( 1974 : 69- - 70) , whe re the defective upp er est ima te for a n is replaced by a val id o ne.

A I = 0 . " ( I t a l i c s i n t h e o r i g i n a l . ) B O R E L ( 1 9 0 9 : 2 5 0 ) .

W h y is A 1 = 0 a c o n s e q u e n c e o f l ir a a , = 0 ? P r e s u m a b l y b e c a u s e

A 1 = l i m a , .n~oo

L e t u s i n t r o d u c e s o m e n o t a t i o n l a c k i n g i n B O R E L'S p a p e r : w e d e n o t e b y E z t h e s et

o f s e q u e n c e s h a v i n g e x a c t l y o n e s u cc e ss , a n d b y E ~ th e s e t o f s e q u e n c e s w i t h e x a c t l y

o n e s u c c e s s a m o n g t h e f i rs t n o u t c o m e s . T h e r e l a t i o n b e t w e e n t h e s e ts E I a n d t h e

s e t s E ] i s t h a t , a s s e t s ,

E 1 = l i m E l .

I t f o l l o w s , by countable additivity, t h a t

A 1 = P ( E 1 ) = l ir a P (E ] )= l im ~r ,. 1n~o3 n~oo

T h e f a c t t h a t B O R E L c o u l d i m a g i n e h e h a d f o u n d a n alternative a r g u m e n t t oo3

e s t a b li s h A 1 = ~ c o , i n t hi s f a s h i o n o n l y r e in f o r c es t h e i m p r e s s i o n t h a t h e n e i t h e r

f u ll y re c o r g n i z e d w h e n h e w a s e m p l o y i n g c o u n t a b l e a d d i t iv i t y n o r a p p r e c i a t e d i ts

p r i m a c y i n l i m i t r e l a t i o n s .

T h e c a s e s A 2 a n d t h e n A k a r e t r e a t e d s i m i l a r ly . B O R E L w r i t e s

Ak = Ao ~ uil ... ui~

a n d n o t

A~ = ~ (Ao u ~ , . . . u ~ ) .

T h i s g a v e ri se t o t h e s a m e m i s g i v i n g s a b o u t t h e d i v e r g e n t c a s e q u o t e d a b o v e i n

c o n n e c t i o n w i t h ( 4 . 3 ) a n d ( 4 . 4 ) .

4.4. The Calculation of Ao3

T h e d e r i v a t i o n o f A o3 f u r n i s h e s s t i ll m o r e e v i d e n c e o f B O R E L 'S r e l u c t a n c e t o r e l y

f u l ly o n c o u n t a b l e a d d i t i v i t y . F i r s t, i n t h e c o n v e r g e n t c a s e , e a c h A k is p o s it i v e , k

= 1 , 2 , 3 , . . . , a n d i f w e d e f i n e S b y

t h e n

S = A o + A 1 + A 2 + " ' + A k + ' " ,

S = A o ( 1 + Z u f l + Z u i ~ u i 2 + . . . + Z u h u i 2 . . , u i + . . . )

= A o ( 1 + u l )( 1 + u 2 ) . . . (1 + u , ) . . . .

1 Theorem s evaluating lira P(E,)under v arious circumstances were the m ain results in the periodn~co

before BOREL 1909). BERNOULLI'S "W eak") Law of Large N umb ers, an d the Central Limit Theoremare theorem s of this type. They did n ot assert th at this lim it was itself the probab ility o f any single event,howe ver, wh ereas BOREL'S heorem did.On this im portant point, see the discussion in § 5.3 of BOREL'S("Strong") L aw o f Large Num bers.

I t is t h is p r o d u c t f o r S ( r a t h e r t h a n t h e d e f i n i n g s u m ) w h i c h is t h e n e m p l o y e d . S i n c e

i t f o l l o w s t h a t

U n - - l _ _ p n '

1l + u . -

1 - p . "

T h u s t h e i n f i n i t e p r o d u c t i n t h e e x p r e s s i o n f o r S c a n b e e v a l u a t e d b y o b s e r v i n g

[ [ ( l + u . ) = 1 1

I 1 Ao '

s o t h a t

S = I .oo

T h i s i s a p e r f e c t ly r i g o r o u s m a n i p u l a t i o n o f c o n v e r g e n t i n fi n it e p r o d u c t s s i n ce ~ u ,1

c o n v e r g e s. T h e a p p e a l t o c o u n t a b l e i n d e p e n d e n c e o c c u r s i n th e e x p r e s s i o n f o r A o.

T h e a s s e r t i o n t h a t A ~ = I - S " 6 v i d e m e n t " is a c as e o f d i sg u i se d c o u n t a b l e

a d d i ti v it y , p a s s e d o v e r w i t h o u t c o m m e n t .

I n t h e d iv e r g e n t c a se " o n p e u t i n d u i r e " t h a t e a c h A k i s z e r o ; h e n c e a l s o t h e i r s u m

S a n d h e n c e Aoo = 1. T h e r e s u lt is " e x a c t e " , b u t " l e r a i s o n n e m e n t p r d c 6 d e n t m a n q u e

d e r i g u e u r , p o u r d e s r a i s o n s d 6 j / t i n d i q u 6 e s " . B O R E L ( 1 9 0 9 : 2 5 1 ) .

B O R EL t h e r e f o r e c o n s i d e r s a n a l t e r n a t iv e a p p r o a c h , p r e s u m a b l y m o r e r i g o r o u s ,

i n t r o d u c i n g t h e s e t o f s e q u e n c e s w i t h m o r e t h a n m s u c c e s s es a m o n g t h e f i rs t n t r ia l s .

L e t u s c a l l t h i s s e t F 2 , s i n c e B O R E L ' s p a p e r l a c k s a n o t a t i o n f o r it . B O R EL a s s e r t s t h a t

i t i s e a s y t o c a l c u l a t e P ( F 2 ) a n d e a s y t o s h o w t h a t

l i r a P ( F ~ ) = 1 , fo r ev ery m = 1 , 2 , 3 . . . .

H e l e a v e s t h e a r g u m e n t t o t h e r e a d e r . T h e f o l l o w i ng is t h e " o b v i o u s " p r o o f t h a t

p r e s e n t s i t s e l f .

I fE " ~ = t h e s e t o f s e q u e n c e s w i t h e x a c t l y m s u c c e s se s a m o n g t h e f i rs t n o u t c o m e s ,

t h e n b y ( f i n i t e ) a d d i t i v i t y

S i n c e

P ( F 2 , ) + P ( E ~ o) + P ( E " ~ ) + . . . + P ( E T , ) = 1 .

• nh m E k - E k , k = 1 , 2 , 3 , . . . ,

n~ c o

a s s e t s , t h e r e f o r e

l im P ( E ~ ) = P ( E k ) = A k = O ,n~ oo

( b y c o u n t a b l e a d d i t i v i t y ) a n d s o

k = 1 , 2 , 3 , . . .

l i m P ( F ~ ) = 1 , k = 1 , 2 , 3 , . . . .n ~ o o

I f t h is i s t h e a r g u m e n t B O R E L i n t e n d e d t o s u p p l y , i t i s i n d e e d a n e a s y o n e , b u t n o

i m p r o v e m e n t w i t h r e sp e c t t o " l a c k o f r i g o r " i n a s m u c h a s t h e u n d e r l y i n g l im i t

r e l a ti o n s a r e n o t e s t a b l is h e d . P e r h a p s B O R EL i n t r o d u c e d t h e s e ts w h i c h w e h a v e

c a l le d F ~ b e c a u s e t h e y a r e d e f i n e d b y e x p l i c it c o n d i t io n s o n t h e f ir st n o u t c o m e s (i.e.,

t h e y a r e " c y l i n d e r s e t s " in t h e p r o d u c t s p a ce ) . I f s o , th i s i n c r e a s e i n e x p l i c i tn e s s is

n o t a c c o m p a n i e d b y a n y i n c r ea s e i n r ig o r : a t s o m e p o i n t a li m i t o f p r o b a b i l it i e s,

s u c h a s l i m P(FT,) , i s c o m p u t e d . S u c h a l i m i t c a n b e i d e n t i f ie d a s b e i n g i t s e l f an ~ o o

p r o b a b i l i t y o n l y b y e m p l o y i n g c o u n t a b l e a d d i t i v i t y . T h a t i s ,

(as se t s ) and

• nh m F.~ = F, .n ~ o o

l im P(F2, = P( l i m F2)n ~ o o n - - ~ o o

( b y c o u n t a b l e a d d i t iv i t y ) to g e t h e r y i e l d

P(Fm)= l i m P( F~) .

T h e t o t a l a b s e n c e o f s et n o t a t io n a n d s et a lg e b r a c a n o n l y h a v e m a d e i t m o r e

d i f f i cu l t f o r BO REL t O f o r m ul a t e , i n a c l ea r w ay , t he c r i t i c a l r e l a t i on

lira/7,~ = F , , .

B O RE L e m p h a s i z e d t h e s i g n i fi c an c e o f t h e c o n c l u s i o n , w h i c h i n o u r n o t a t i o n is,

l i ra P(F, ] )= 1:t l ~ o o

. .. ; t h is m e a n s t h a t o n e c a n p r o f i t a b l y b e t o n e f r a n c a g a i n s t a r b i t r a r i ly l a rg e

o d d s t h a t t h e n u m b e r o f s u c c es s es w i ll e x c e e d a n y f i xe d n u m b e r m ; th i s is th e

p r e c is e m e a n i n g o f t h e a s s e r t i o n t h e p r o b a b i l i t y A ~ i s o n e. BOREL (1909: 251) .

4 . 5 . A d d i t i o n a l E v i d e n c e o n t h e P r i m a c y o f C o u n t a b l e I n d e p e n d e n c e

B e f o r e c o n c l u d i n g h i s C h a p t e r I , B O R EL c o n s i d e r s v a r i o u s g e n e r a l i z a t io n s

a n d m o d i f i c a ti o n s o f t h e a b o v e m a i n t h e o r e m c o n c e r n i n g A 0 , A 1 , A 2 , . .. , a n d A ~

i n t h e c o n v e r g e n t c a se . A s a p r e l i m i n a r y h e c o n s i d e r s t h e c a s e o f a s i n g le t r i a l in

w h i c h t h e r e is a d e n u m e r a b l e i n f in i ty o f o u t c o m e s . H e a l so c o n s i d e r s a n i n f in i te

s e q u e n c e o f s u c h t r ia ls , a s s u m e d i n d e p e n d e n t , t h e p r o b a b i l i t y o f t h e n th o u t c o m e

o n t h e d h t r i a l d e n o t e d P , ,s . B O R E L t h e n a s s u m e s

~ p , , s = l f or s = 1 , 2 , 3 . . . .n = l

w h i c h m e a n s t h e o u t c o m e s a r e e x h a u s t i v e i n e a c h t r i a l . H e f u r t h e r r e s t r i c t s

c o n s i d e r a t i o n t o t h e " f u l l y c o n v e r g e n t " c as e, m e a n i n g t h a t ~ P,,s c o n v e r g e s f o r

n = 1 , 2 , 3 , . . . , s = l

T h o u g h t h e t r ia l s a r e a s s u m e d i n d e p e n d e n t , t w o e v e n t s d e t e r m i n e d b y th e

n u m b e r o f o c c u r r e n c e s o f t w o d i ff e re n t o u t c o m e s n e e d n o t b e i n d e p e n d e n t .

14 0 J. BARONE & A. NOVIKOFF

L e t u s i n t r o d u c e t h e f o l l o w i n g n o t a t i o n :

E , , k = s e t o f t ri a l s eq u e n c e s f o r w h i c h t h e n th o u t c o m e o c c u r s e x a c t l y k t i m e s ,

E , , ~o = s e t o f tr i a l s e q u e n c e s f o r w h i c h t h e n th o u t c o m e o c c u r s i n f i n i te l y o f t e n ,

E , ,k = s e t o f t r i a l s e q u e n c e s f o r w h i c h t h e n th o u t c o m e o c c u r s a t m o s t k ti m e s .

I f w e c o n s i d e r a n y o n e i n d i v i d u a l o u t c o m e , s a y t h e n th, i t f o l lo w s f r o m t h e

" c o n v e r g e n c e " h y p o t h e s i s t h a t

P ( E n , ~ ) = 0

o r , e q u i v a l e n t l y ,

P ( G , o o )= 1.

N o w w e e n c o u n t e r a n e x a m p l e o f w h a t w e h a v e c al le d " l i m i t e d " c o u n t a b l e

i n d e p e n d e n c e ; a l t h o u g h t h e d if f er e n t o u t c o m e s n e e d n o t i n g e n e r a l b e i n d e p e n d e n t ,

t he eve n t s E c co,, n = 1 , 2 , . . . , h a v i n g p r o b a b i l i t y 1, a r e o f n e c e s si ty m u t u a l l y

i n d e p e n d e n t . I n w o r d s , g i v e n th e " f u l l y c o n v e r g e n t " h y p o t h e s i s , th e r e is p r o b a b i l i t y

1 t h a t a ll o u t c o m e s o c c u r o n l y f in i te l y o ft e n .

I n o u r n o t a t i o n , P ( U , , oo) = 1 fo r n = 1 , 2 , 3 , . . . im pl i es

P E c, oo = I ~ P ( E ~ , ~ ) = 1 . (4 .5 )1

T h i s l i n e o f r e a s o n i n g p r o v o k e s B O R E L t o o n e o f t h e r a r e i n s t a n c e s i n w h i c h

c o u n t a b l e i n d e p e n d e n c e ( e ve n in t h e " l i m i t e d " f o r m ) g iv e s r is e to e x p l ic i t d o u b t s :

I t m a y h e l p t o e x a m i n e t h e q u e s t i o n m o r e c lo s el y, t o b e s u re t h a t o u rr e a s o n i n g i s s t r ic t : t h e f a c t t h a t t h e r e i s a d e n u m e r a b l e i n f in i t y o f f a c t o r s e q u a l t o

1 c o u l d i n d e e d l e a v e s o m e d o u b t a s t o t h e v a l u e o f t h e i r p r o d u c t . B O R EL ( 1 9 09 '

B O R EL t h e n r e a s s u re s h i m s e l f b y p r o v i d i n g a n e l a b o r a t e p r o o f o f a n a s s e rt i o n, i n

t w o p a r t s , w h i c h r e f i n e s a s w e l l a s " r e - e s t a b l i s h e s " t h e r e s u l t i n q u e s t i o n .

F i rs t , h e p r o v i d e s a n a r g u m e n t t h a t f o r e v e r y p o si ti v e i n te g e r k ,

( I n w o r d s , t h e p r o b a b i l i t y t h a t a l l o u t c o m e s o c c u r a t m o s t k t i m e s f o r a n y f i x e d f in i te

i n t e g e r k i s z e r o . ) T h i s i s e s t a b l i s h e d b y a n i n g e n i o u s c a l c u l a t i o n , t h e d e t a i l s o f

w h i c h a r e l e ft t o t h e r e a d e r ( c f B A R O N E (1 9 7 4 : 1 0 1 - 1 1 3 ) f o r a d e t a i l e d d i s c u s s i o n ) ,

t o g e t h e r w i t h t h e t a c i t l y a s s u m e d r e l a t i o n

P & , k = P ( & , O .n 1 n= l

T h i s r e l a ti o n i s t a n t a m o u n t t o a s s u m i n g i n d e p e n d e n c e o f t h e e v e n t s/ ~ , ,k , i . e . , o f

o u t c o m e s , n o t j u s t o f t ri a ls .S e c o n d , g i v e n a n y e > 0 , o n e c a n c h o o s e t h e s e q u e n c e k~ s o t h a t

P(/~,~, j < ek = l

since P(En, oo)=0 for each n. Th en it follows readily, if e<½ , th at

-P (E , , k ,) )= l~I P(/~,,kn) > 1 - 2 e . (4.7)n = l n = l

B O R E L conc ludes :

One can thus, for a given arbitrary e, choose the numb ers k, in such a way that

the proba bili ty that the n th outcom e occurs no more than k, t imes differs fro m unity

by less than 2e. (Italics in th e origin al.) BOR EL (1909: 257).

In o ther words , BOREL is in te rpre t ing the abov e inequa l i ty by aga in us ing

independen ce be tween the events /~ , ,k , th is t ime in the form :

The validi ty of this requires tha t n ot only events def ined by dif ferent t r ia ls but a lso

those def ined by d i f fe rent outcom es a re assumed independent . T hus BOREL has

replaced reasoning based on the ( faul t le ss) pr inc ip le of " l imi ted" countable

independen ce by reasoning which employs a tac i t use of countable independence .

This s econd use suffers f rom the defect tha t i ts applicat io n is val id only subject to an

addi t ion a l hypothes is . (Once aga in , the fact tha t ve rba l conc lus ions abo ut events

a re drawn f rom sym bol ized re la t ions be tween numbers , wi th no in te rvening

notat ion for se ts or events , helps disguise the transgression from a suff ic iently

casual reader , perh aps f rom BOREL himself . )

Thu s BOREL emp loys (4.6) and (4.7) to o vercom e do ubts abo ut the reaso ning of

(4.5) . Of necessi ty these two new asser t ions themselves require countable inde-

pendence and hold only subjec t to an addi t iona l , f a r - reaching (and tac i t )

assu mpt ion, w hereas (4.5) i tse lf can be established with no add it io nal assu mptio ns

bey ond the in i t ia l ly s ta ted ones .

The "ob vio us" p roof of (4 .5), mak ing no appea l wh a tever to independenc e of

outcomes , namely the complemented asse r t ion

P E, ,~ < P(E, ,oo)=0,

is consp icuou sly absent , s ince i t hinges on co unta ble sub -addit ivi ty, never used by

BOREL in the paper of 1909.

This c onclu ding section of BOREL'S Ch apte r I furnishes anot her instance of his

predi lec t ion for avoid ing a rguments based on sub-addi t iv i ty and countable

addi t iv i ty , and for offe ring ins tead a rgu ments based o n countab le independence .

Indeed, this insight into BOREL is the ch ief interest a t t ach ed to this sect ion; i ts

results are nowhere appealed to la ter .

4.6. Summ ary o f Borel ' s C once ptual S hortcom ings in Ch apter I

In sum mary, BOREL fai ls to em ploy sub-addi t iv i ty and seems to d ou bt rout ine

arguments involv ing countable addi t iv i ty , of fe r ing a l te rna t ives which employ

142 J. BARONE~z A . NOVIKOFF

s o p h i s t i c a te d r e f o r m u l a t i o n s o f c o u n t a b l e a d d i t i v i t y s u c h a s

P ( l i m E , ) = l i ra P ( E . ) ;

h e b a s e s h i s r es u lt s, s o m e t i m e s n e e d l es s ly , o n c o u n t a b l e i n d e p e n d e n c e a n d a s s u m e s

t h e m v a l i d i n c i r c u m s t a n c e s w h e r e n o i n d e p e n d e n c e h a s b e e n a s s u m e d o r

e s t a b li s h e d . C o u n t a b l e i n d e p e n d e n c e i t s e lf is t a k e n a s b e in g e v i d e n t b y a n a l o g y

w i t h t h e f i n i t e c a s e d e s p i t e o c c a s i o n a l r e s e r v a t i o n s . N o n o t a t i o n i s i n t r o d u c e d f o r

s et s, a n d h e n c e n o n e f o r se t fu n c t i o n s . T h e a d j e ct i ve " d e n u m e r a b l e " , a p p r o p r i a t e

f o r th e n u m b e r o f t ri a ls , is o c c a s i o n a l l y a n d e r r o n e o u s l y c o n s t r u e d t o r e fe r to t h e

s iz e o f t h e s a m p l e s p a c e , a n d i t i s as s e r te d t h a t t h e t h e o r y o f " d e n u m e r a b l e

p r o b a b i l i t i e s " is a m o r e " e f f e c t iv e " t h e o r y t h a n t h a t o f t h e c o n t i n u u m , s i n ce th e

l a t te r c l a im s t o t r e a t o f m o r e t h a n d e n u m e r a b l y m a n y e l e m e n t s a s b e i n g o n e

c o l l e c t i o n . ( C f t h e o p e n i n g r e m a r k a n d c o n c l u d i n g l i n es o f BO R EL 's p ap e r , b o t hg iven in § 4 .1 .) Th i s l a s t d i s t in c t io n i s o f cours e i l l u so ry .

H a d t h e p a p e r c o n t a i n e d n o m o r e , it w o u l d h a v e f u r n i s h e d n o e v i d e n c e

w h a t e v e r t o f a v o r t h e h y p o t h e s i s t h a t B O R E L u n d e r s t o o d t h a t h i s " p r o b a b i l i t 6 s

d 6 n o m b r a b l e s " w a s a f o r m o f m e a s u r e t h e o r y ( in p a r t i c u l a r t h a t c o u n t a b l e

a d d i t i v i t y w a s a n i n t ri n s ic p a r t o f i ts a p p a r a t u s ), a n d a ll t h e a f o r e m e n t i o n e d t o

d i s p u t e it . H o w e v e r , t h e t w o r e m a i n i n g s e c t i o ns o f t h e p a p e r s o m e w h a t c o u n t e r b a l -

a n c e t h i s o v e r - s i m p l e i n t e r p r e t a t i o n .

5. Borel's Chapter II: The Strong Law of Large N umbers

5 .1 . The Se t t ing o f the Prob lem

C o n s i d e r t h e d e c i m a l e x p a n s i o n ~ ( b n / 1 0 " ), e a c h b~ b e i n g o n e o f t h e d i g i t s

0 , 1 , . . . , 9 . T h e m o r e g e n e r a l p r o b l e m o f " q - a r y " e x p a n s i o n s , ( b f f q ~ ) ,w h e r e e a c hn=l \

b , is a m o n g t h e i n t e g e r s 0, 1 , . . . , q - 1 , c a n o f c o u r s e b e t r e a t e d i n l i k e m a n n e r . )/

B O RE L p r o p o s e d t o s t u d y t h e p r o b a b i l i t y t h a t s u c h b n " b e l o n g t o a g i v e n s e t "

a s s u m i n g ( 1 ) t h e d i g i t s a r e i n d e p e n d e n t a n d , ( 2 ) e a c h d i g i t h a s e q u a l p r o b a b i l i t y(na m e ly 1 /10 ) o f a ch ie v ing the va lues 0 , 1, . . . , 9 .

I t is n o t n e c e s s a r y to e m p h a s i z e t h e p a r t l y a r b i t r a r y n a t u r e o f t h e s e tw o

h y p o t h e s e s : t h e f i rs t, i n p a r t i c u l a r , i s n e c e s s a r i l y i n a c c u r a t e w h e n o n e c o n s i d e r s ,

as one i f f o r ce d to in prac t i ce , t h a t a d e c i m a l e x p a n s i o n i s d e f i n e d b y a law,

w h a t e v e r m i g h t b e t h e n a t u r e o f t h a t l aw . I t m a y n o n e t h e l e s s b e in t e r e s t in g t o

s t u d y t h e c o n s e q u e n c e s o f t h i s h y p o t h e s i s , p r e c is e l y i n o r d e r t o r e a l iz e t h e e x t e n t

t o w h i c h t h i n g s o c c u r as i f t h i s h y p o t h e s i s w e r e v e r if ie d . T h e s e c o n d h y p o t h e s i s ,

t h a t i s t h e e q u a l i t y o f p r o b a b i l i t i e s f o r t h e v a r i o u s p o s s i b l e v a l u e s o f e a c h d e c i -

m a l d i g i t , s e e m s r a t h e r n a t u r a l , g r a n t i n g t h e f i r s t .

T h e s e t w o h y p o t h e s e s a r e e a s i ly j u s t if i e d a d d i t i o n a l l y b y t a k i n g n o t t h e

l o g i c a l, b u t t h e g e o m e t r i c p o i n t o f v i e w : t h e y a r e, i n d e e d , e q u i v a l e n t t o t h e

f o l l o w i n g : t he dec ima l number be ing r epre sen ted by a po in t o f the in t e rva l [0, 1],

the probabi l i ty that i t i s located in a subinterval i s equal to the length o f that

A x i o m a t i c P r o b a b i l i t y 1 4 3

subinterval . O n e c o u l d i n t e r p r e t a n d v e r if y t h e r e s u lt s w e a r e g o i n g t o o b t a i n

f r o m t h is g e o m e t r i c p o i n t o f v i e w ; I w il l n o t d o s o , p r e f e r r i n g t o l e a v e a s i d e f o r

t h e p r e s e n t t h e t h e o r y o f c o n t i n u o u s p r o b a b i l i t y , w h i c h i s c o n n e c t e d , a s I h a v e

s h o w n e l s e w h e r e , t o t h e t h e o r y o f m e a s u r e o f s et s. 1 B O R E L (1 9 0 9 : 2 58 ).

T h u s B O R EL 'S " c o n s t r u c t i v i s m " l e a d s h i m t o a s s e r t t h a t t h e f ir st h y p o t h e s i s i s

" n 6 c e s s a i r e m e n t i n e x a c t e ' . O n e is l e d t o s p e c u l a t e t h a t B O RE L k n e w w h e r e h e

w a n t e d t o g o a n d a r r a n g e d t o g e t t h e re , a t t h e e x p e n s e o f hi s s c ru p l e s i f n e c e s s ar y .

T h e r e m a r k a b o u t t h e s e c o n d h y p o t h e s i s i s , p e r h a p s , e v e n m o r e i n t e r e s t i n g :

n o t h i n g i n t h e f i r s t c h a p t e r ( " p o i n t d e v u e l o g i q u e " ) d e m a n d s e q u a l p r o b a b i l i t i e s

1/q f o r e a c h b n. T h e g e n e r a l t h e o r y o f tr ia ls , e a c h w i t h q o u t c o m e s , a c c e p t s a n y

c h o i ce s p . . .. n = 0 , 1 . . . . q - l , s u c h t h a t

p o , s + p l , s + . . . + p q _ l , s = l , s = 1 , 2 , 3 , . . . .

5 .2 . The Spec ia l Case p , = 1 fo r Dyad ic Exp ans ions

B O R E L r e a l iz e s e x p l i c i tl y t h a t t h e s e c o n d h y p o t h e s i s o f e q u a l l y l i k e ly b , p e r m i t s

p r o b a b i l i t y t o b e i n t e r p r e t e d t w o e q u i v a l e n t w a y s , o n e o f w h i c h i s f a m i li a r. I n t h e

c a s e q = 2 , t h is m e a n s t h e p r o b a b i l i t y o f t h e d i g i t 0 a n d o f t h e d i g i t i i n t h e n t h p l a c e

a r e n o t o n l y i n d e p e n d e n t o f t h e c h o i c e o f di g it s in t h e o t h e r p l a c e s b u t a r e b o t h 1 . ( I t

is a n i n t e r e s ti n g c o m m e n t o n t h e c h a r a c t e r o f m a t h e m a t i c a l e v o l u t i o n t h a t b e f o r e

S TE IN H A U S ( 19 23 ) n o o n e h a d t h e t e m e r i t y to c o n s i d e r w h e t h e r a n y o t h e r c h o i c e o fp r e - a s s i g n e d p r o b a b i l i t i e s { P n} d e p e n d i n g i n g e n e r a l o n n c o u l d a l s o i n d u c e a

m e a s u r e o n [ 0 , 1 ] . 2) B O R E L n o w s t u d i e d t h i s e q u i p r o b a b l e c a s e b y h i s n e w

p r o b a b i l i s t i c m e t h o d s i n p r e f e r e n c e t o m e a s u r e - t h e o r e t i c o n e s ( al so h is o w n

i n v e n t i o n ) .

P r o c e e d i n g t o t h e c a s e o f o n l y t w o o u t c o m e s f o r e a c h t ri a l, t h e c a s e q = 2 , w i t h

t h e c o n v e n t i o n t h a t t h e d i g i t 0 i s a s u c c e s s , o r f a v o r a b l e c a s e , B O R E L s t a t e s

O n e k n o w s t h a t , if o n e c o n s i d e r s 2 n t ri al s, t h e p r o b a b i l i t y t h a t t h e n u m b e r o f

f a v o r a b l e c a s e s w i l l l i e b e t w e e n

is eq ua l to 0( ,~) , l e t t in g

n - , ; L ~ n a n d n ÷ ) d f f n

BOREL (1909: 259) .

1 T h i s i s th e s o l e r e f e r e n c e t o t h e t h e o r y o f m e a s u r e i n t h e e n t i r e p a p e r o f 1 90 9. I t s r o l e i s t o n o t i f y t h e

r e a d e r t h a t t h e t h e o r y o f m e a s u r e , a n a l t e r n a t i v e a p p r o a c h , i s no t b e i n g e m p l o y e d . B O R E L c l e a r l y i s

r e f e r r i n g t o h i s p a p e r o f 1 9 0 5 t h e c o n t e n t s o f w h i c h w i ll b e d i s c u s s e d i n § 7 .3 .

2 T h e s t u d y o f t h e s e m e a s u r e s , i m p l i c i t ly i n t r o d u c e d b y B O R E L, w a s f i r s t e x p l o r e d i n a n y d e t a i l a s

r e c e n t l y a s 1 9 4 7 (WINTNER) n d 1 9 4 8 (H A R TM A N ). T h e n o t i o n o f g e n e r a l i z i n g B O R R ( o r L EB ES GU E)

m e a s u r e t o o t h e r c o u n t a b l y a d d i t iv e s e t fu n c t i o n s h a d o c c u r r e d w i t h i n i n t e g r a t io n t h e o r y i n 1 91 3

( R A D 6 N ) a n d w i t h i n m e a s u r e t h e o r y i n 1 9 1 4 ( CA R ~r H~ O DO a Y) .

( T h e s t a t e m e n t i s o f c o u r s e u n t r u e , b u t t h e l imit o f t h i s p r o b a b i l i t y , a s n t e n d s t o

i n f i n i t y , i s g i ven by 0 ( 2 ) . ) As s um i ng

l i ra ~nn = 0n ~ o o

b u t 2 , i t s e l f g r o w s u n b o u n d e d l y , B O R E L c o n s i d e r s e a c h s e t o f 2 n i n i ti a l tr i a ls a s

d e t e r m i n i n g a n e v e n t a s f o ll o w s . L e t v,(x) b e t h e n u m b e r o f z e r o s i n t h e f i rs t n d i g it s

o f t h e b i n a r y e x p a n s i o n o f x . F o r e a c h n , t h e " t r i a l " e x a m i n e s v2,(x) , a n d t h e r e s u l t is

a " s u c c e s s " i f

I v z , ( X ) - n l > 2 , 1 f n

a n d a " f a i l u r e " i f

Ivz,(x)-nl <& I/L

T h e p r o b a b i l i t y p , o f a f a v o r a b l e c as e is a ss e rt e d t o b e

2 ~ e_~2d2

a n d t h e p r o b a b i l i t y q , o f a n u n f a v o r a b l e c a s e is g i ve n b y q , = 1 - p ,. 1

B O R E L n o w focuses h is a t t e n t i o n o n t h e s et E ~ o f t h o s e " d y a d i c e x p a n s i o n s " f o rw h i c h i n f in i t el y m a n y " t r i a l s " h a v e " s u c c e s s e s " . A l w a y s f a s c in a t e d b y s e ts o f r e a l

n u m b e r s c h a r a c t e r iz e d b y a d e n u m e r a b l e s et o f c o n d i ti o n s , B OR EL h a d ju s t f o u n d an

n e w s e t, w i th a n e w " d e n u m e r a b l e " d e s c r ip t io n . F u r t h e r m o r e , s i nc e ~ , p , c o n v e r g e s1

i f 2 , g r o w s s u f f i c i e n t l y f a s t (e .g . n~ ), h i s Z e r o - O n e L a w a p p l i e d t o t h is c a s e ( a s s u m i n g

t h e v a l id i t y o f i ts a p p l i c a t io n ) a s s e r ts t h a t A ~ = 0 a n d s o P ( E ~ ) = 1 .

1 This ca lculation of p. and q. is serious ly flawed. Wha t is true is that p. and

2 ~ e_a~ d2

are of the same o rder, b ut since ~.. is not fixed, this is not a case of the classical Central Limit Theorem.

The information needed, that

is a refined and relatively recent result. Without it the convergence of

(which is true if2. grows rapidly enough, e.g., n+) need not imply the convergence of~, p.. However, even

assumin g }~p. converges, one cannot conclude f rom BOREL'S result for A~ that there is a zero

probability of infinitely ma ny un favorable cases, since that result was established on the hypothesis that

the separate trials be independent. The various "trial s", with probability p. a nd q. o f success and failure

as defined above, are by no means independent.

To exam ine the se t E~ in mor e de tai l , no te tha t i f the n th t r i a l is unf avo rab le the

r a t i o o f th e n u m b e r o f 0 's to t h e n u m b e r o f l ' s a m o n g t h e f ir st n d i gi ts l ie s b e t we e n

or , equ iva len t ly , be tween

I f the r e c an b e on ly f in i te ly m an y exce pt ions to th i s, i.e., i f there are o nly f ini te ly

m an y "successes" , the r a t io o f 0 's to l ' s mu s t t en d to the l imi t 1 a s n t ends to in f in i ty .

T h u s t h e s et E ~ c o n s is t s o f t h o s e n u m b e r s x wh o s e d y a d i c e x p a n s i o n s h a v e

a s y m p t o t i c a l l y t h e s a m e n u m b e r o f z e r o s as o ne s . S in c e P ( E o o ) = A ~ = 0 , t h e s e

nu mb er s cons t i tu te a set o f p ro bab i l i ty 1. This i s the BOREL Law of l a rge Nu mb er s .

5.3. Normal Numbers

T h e c o r r e s p o n d i n g r e a s o n i n g a p p l i e s t o t h o s e n u m b e r s w i t h i n wh o s e d e c i m a l

expa ns ion s the d ig i t s 0, 1 , 2 . . . . . 9 each have a l imi t ing f r equency o f 1/10 : they fo rm a

se t o f p rob abi l i ty 1. Thes e nu mb er s BOREL ca l led "simply normal" to the base 10.

M or e g enera l ly , BOREL ca l led a num be r s im ply nor ma l to the base q if the d ig i t s in

t h e q - a r y e x p a n s i o n

f o r i = 0 , 1, . . . , q - 1 .

F o r e a c h q t h e s et o f n u m b e r s s i m p l y n o r m a l t o t h e b a s e q is s i m i l a rl y s e en t o b e

of p rob abi l i ty 1. A n um be r i s ca l led entirely no rm al to the base q if i t i s s imply

n o r m a l t o e a c h o f t h e b a s e s q, q2 .... , qk, .... A number i s ca l led absolutely n o r m a l if

i t i s s imp ly norm al to the base q fo r eve ry q = 2 , 3 . . . . I f we deno te the se t o f nu mb er s

s i m p l y n o r m a l t o t h e b a s e q b y Nq, t h e n t h e s e t o f a b s o l u t e l y n o r m a l n u m b e r s is

a n d t h e s et o f n u m b e r s e n t i re l y n o r m a l t o t h e b a s e q is

146 J. BARONE& A . NOVIKOVF

B O R E L a s se r ts ( w i t h o u t a r g u m e n t ) t h a t b o t h t y p es o f n u m b e r s a r e o f p r o b a b i l i t y 1.

W h a t is b e i n g t a c i t l y a f f i r m e d i s, a s u s u a l, t h a t

a n d t h a t

P = P ( N q ~ )k= l

w h e r e e a c h f a c t o r o n t h e r i g h t - h a n d s i d e is 1. T h e s e a r e c a se s o f " l i m i t e d " c o u n t a b l e

i n d e p e n d e n c e . A l a c o n i c f o o t n o t e r e a ds

I d o n o t t h i n k i t s e rv e s a n y p u r p o s e t o r e p e a t t h e d e t a i le d p r o o f o f t h e f a c t

t h a t o n e h a s t h e ri g h t t o a p p l y t h e t h e o r e m o f c o m p o s i t e p r o b a b i l it i e s , d e s p i t et h e d e n u m e r a b l e i n f in i ty o f c as e s. B O R g L ( 1 9 0 9: 2 6 1 : F o o t n o t e (5)).

O f c o ur s e, n o " d e t a i l e d p r o o f o f t h e r ig h t t o a p p l y t h e t h e o r e m o f c o m p o u n d

p r o b a b i l i t i e s " i n t h e d e n u m e r a b l y i n fi n it e c a s e h a s b e e n g i v e n ; r e ca l l th a t i t w a s fi rs t

m e n t i o n e d i n c o n n e c t i o n w i t h

A 0 = ( 1 - p l ) ( 1 - P 2 ) . . . (1 - p , ) . . .

" D a n s le c a s d e la c o n v e r g e n c e , l ' e x t e n s i o n d u p r i n c i p e d e s p ro b a b i l i t 6 s

c o m p o s 6 e s v a d e s o i, . . . " B O R E L (1 9 0 9 : 2 49 ). R e p e a t e d u s e h a s b y t h i s p o i n t

t r a n s f o r m e d t h e p r in c i p l e f r o m a s e lf - e v id e n t t r u t h t o o n e w h o s e p r o o f h a s b e e nd e m o n s t r a t e d a ll t o o o f te n .1

B O R E L ' s r e a s o n i n g , u p t o t h i s p o i n t , i s c h i e f ly f l a w e d i n th e f o l l o w i n g t w o w a y s :

h e e m p l o y s a r e su l t b a s e d o n i n d e p e n d e n c e o f e v e n t s f o r d e p e n d e n t t ri a ls ( th is f la w

w o u l d b e r e m e d i e d b y a " C A N T E L L I " m o d i f i c a t i o n o f h is Z e r o - O n e L a w , c f § 6.4)

a n d h e u s es a f o r m o f t h e C e n t r a l L i m i t T h e o r e m m o r e p r e c is e t h a n w a s t h e n

a v a il a b le . H A U S D O R F F ' s a n d l a t e r p r o o f s w e r e t o a v o i d a n y a p p e a l t o t h e C e n t r a l

L i m i t T h e o r e m w h a t e v e r .

A n a d d i t i o n a l o m i s s i o n is n o t e w o r t h y : d e s pi te a c h o i c e o f n o t a t i o n i d e nt i ca l t o

t h e o n e h e h a d e m p l o y e d f o r d is c u s si n g t h e (B E R N O U LL I) L a w o f L a r g e N u m b e r s i n

h i s t e x t - b o o k o n P r o b a b i l i t y , w r i t t e n e a r li e r i n t h e s a m e y e a r ( BO R E L ( 1 90 9 a : 6 3 -65)), h e m a k e s n o a t t e m p t t o c o m p a r e t h e B E R N O U L L I L a w a n d t h e n e w r es u lt . A

g r e a t o p p o r t u n i t y is l o st t h e r e b y : t r e a t i n g b o t h w i t h in t h e t h e o r y o f m e a s u r e , o n e

w o u l d h a v e b e e n le d t o th e c o m p a r i s o n o f c o n v e r g e n c e " i n m e a s u r e " a n d

c o n v e r g e n c e " a l m o s t e v e r y w h e r e " , a n t i c i p a t i n g t h e t r e a t m e n t s o f S LU T SK Y ( 19 25 ),

F RI~ CH ET (1 9 3 0) a n d t h e e a r l i e r s u c h c o m p a r i s o n s b y C A N T E L L I ( 1 91 7 ) a n d P O L Y A

( 19 21 ). I t is p o s si b l e t h a t t h e s t r o n g e r c h a r a c t e r o f h i s r e su l t b y c o m p a r i s o n w i t h

1 In discussing A0, Ak, A~ independence of trials was assumed. The independence o f the eventsNq, q = 2, 3,..., or o f he events Nqk, k = 1, 2, 3,..., is slightly mo re sophisticated. It is no t to be regarded asan additional ad hoc assumption. Rather one must establish the general fact that if P ( N ) = 1, then

P ( N c ~ A ) = P ( N ) P ( A ) . i .e ., N is independent of a n y other eve nt A. This BORE5 failed to d o.The proo f involves considering the complement of N c~A and then concluding that a set contained

in a set of probability zero m ust itself have probability zero. The form er consideration has the p ossibledrawback, for BOREL,of shifting attention away from "independenc e". As to the latter, BOREL s knownto have rejected the corresponding reasoning w hen employed in the context o f measure theory.

Ax ioma tic Probab ility 147

B E RN O U LL I'S w a s s u f f i c ie n t ly e v i d e n t t o B O R EL t h a t n o c o m m e n t w a s f o r t h c o m i n g .

B o t h C A N T E L L I a n d P OL Y A w e r e t o b e e x p l ic i t a b o u t t h e d i s t i n c t i o n b e t w e e n t h e

S t r o n g L a w a n d B E R N O U L L I ' S r e s u l t .

I n d e e d , t h e e a r l i e r s u c c e s s o r s o f BO R EL m a y b e c l a s si fi e d r o u g h l y a s b e i n g

" P r o b a b i l i t y " o r ie n t ed o r " M e a s u r e T h e o r y " o r ie n te d . T h e f o r m e r m a y i n v a r ia b l y

b e d i s t i n g u i s h e d b y th e i r i n t e r p r e t a t i o n o f B O R E L 's S t r o n g L a w a s a n a s t o n i s h i n g

r e f i n e m e n t o f B ER N OU L LI'S o r i g i n a l t h e o r e m , w h e r e a s t h e l a t t e r f a i l t o a s s o c i a t e t h e

t w o , s in c e B E R N O U L L I's t h e o r e m s e e m e d n o t t o b e l o n g t o m e a s u r e t h e o r y .

5 .4 . A P o s s i b l e C l u e t o th e G e n e s i s o f B o r e l ' s S t r o n g l a w

T h e r e s u l t o f BO R EL 'S C h a p t e r I I , n a m e l y , B O R E L 's S t r o n g L a w o f L a r g e

N u m b e r s , h a s p r o v e d t o b e a n e x c e e d i n g l y f r u it fu l a p p l i c a t i o n o f t h e A oo = 0 r e s u lt

o f C h a p t e r I , t h e " g e n e r a l " t h e o r y o f d e n u m e r a b l e p r o b a b i li ti e s . T h e r e r e m a i n s t h e

f a s c i n a t i n g q u e s t i o n , w h a t d r e w B O RE L'S a t t e n t i o n t o t h e i n s t a n c e s o f h i s g e n e r a l

t h e o r y f u r n is h e d b y d i g i ts o c c u r r i n g i n t h e d y a d i c e x p a n s i o n o f n u m b e r s i n t h e u n i t

i n t e r v a l? O n e p o s s i b l e a n s w e r i s, o f c o u rs e , B O R E L 's p r o f o u n d i n t u i ti o n . I t is o f

i n t e r e s t to n o t e , h o w e v e r , t h a t i n 1 9 0 8 th e A m e r i c a n , E . V AN VLECK , h a d b e e n l e d to

s t u d y t h e s e t o f d y a d i c i r r a t io n a l s i n t h e u n i t i n t e r v a l d e f i n ed , i n th e n o t a t i o n

i n t r o d u c e d a b o v e , b y t h e r e l a t i o n

l i m v ~( x) _ ~ n - v n ( x )

o r , e q u i v a l e n t l y ,~ . ( x ) / n - - 1 - v . ( x ) / n

l i m = l i m,~ oo 1 - v , ( x ) / n ,4 00 v , ( x ) / n

L e t u s d e n o t e t h i s s e t b y V0 . B O R E L ' s r e s u l t, i f s t a t e d p u r e l y i n t e r m s o f m e a s u r e

t h e o r y , i s t h a t t h e s e t B o d e f i n e d b y

l im v n ( x ) - 1

n ~ n 2

is o f m e a s u r e 1. I t is i m m e d i a t e f r o m t h e i r d e f i n i t i o n s t h a t B o c V0 . O f g r e a t e s t

i n t e r e s t i n th i s c o n n e c t i o n is VAN V L EC K 'S e x p l i c it a n d p i v o t a l c o n j e c t u r e t h a t Vo i s

n o t a s e t o f m e a s u r e 1 ; V A N V L EC K w a s l e d t o t h e c o n j e c t u r e b y a g e n e r a l c r i t e r i o n

f o r n o n m e a s u r a b i l i t y o f s e ts in t h e u n i t i n t e r v a l. I f Vo w e r e n o t m e a s u r a b l e , o r i f i t

w e r e t o h a v e L E B ES GU E in n e r m e a s u r e l e s s t h a n o n e , t h e n V A N V L E C K w o u l d h a v e

c o n s t r u c t e d a n o n m e a s u r a b l e s e t i n t h e u n it i n t e rv a l w i t h o u t u s i n g th e A x i o m o f

C h o i c e ; t h e s e t c o n s t r u c t e d ( ju s t t h e s e t Vo d e f i n e d a b o v e ) w o u l d h a v e b e e n t h e s e t o f

d y a d i c i r r a t i o n a l s d e f i n e d b y

l im vn(x ) /n ~ 1 - v~ (x ) /n

,~ oo 1 - v , ( x ) / n , 4 00 v , ( x ) / n

V A N V L EC K 'S p a p e r c o n t a i n s m u c h e ls e o f i n t e r e s t a n d is d is c u s s e d i n d e t a i l i n

N O V I K O F F & B A RO N E (1 97 7). T h e r e i t is s h o w n t h a t a n e l e m e n t a r y o b s e r v a t i o n

( w h i c h VAN V L EC K u n a c c o u n t a b l y f a il e d t o m a k e ) s h o w s a t o n c e t h a t Vo is

m e a s u r a b l e a n d i n d e ed , f r o m VA N VLECK ' S o t h e r r e s u l t s , m u s t h a v e m e a s u r e 1. V A N

1 4 8 J . BA RO N E & A . N O V IK O FF

V LE CK h a d m a d e B O RE L's a c q u a i n t a n c e i n F r a n c e i n N o v e m b e r 1 9 0 5 - J a n u a r y

1 90 6, w h i l e o n s a b b a t i c a l l e av e , a n d t h e t w o r e m a i n e d o n f r ie n d l y t e r m s f o r y e a r s

t h e r e a f t e r . H a d h e c o m m u n i c a t e d h i s r e s u l t s ( i n c l u d i n g h i s c o n j e c t u r e a b o u t t h e

m e a s u r e o f V 0) t o B O R E L a t o n c e , i t is p o s s i b l e t h a t B O R E L w o u l d h a v e s e e n t h a t t h e

q u e s t i o n o f th e p r o b a b i l i t y ( n o t " m e a s u r e " ) o f V , a n d i n d e e d , t h e m o r e d e l i c a t e

q u e s t i o n o f i ts s u b s e t B 0 , w a s a c c e s s i b le t o h is n e w t h e o r y o f " d e n u m e r a b l e

p r o b a b i l i t y " . W h e n o n e c o n s i d e r s t h a t V A N V L EC K 'S m a i n g o a l w a s t o c o n s t r u c t a

n o n m e a s u r a b l e s e t ( th e v e r y e x i s t e n c e o f w h i c h w a s d i s c o m f o r t i n g to BOREL), h e

o p p o r t u n i t y o f f e r e d B O R EL t o g e t a b e t t e r r e s u l t b y h i s " m o r e e f f e c ti v e " t h e o r y

m i g h t w e l l h a v e d i r e c t e d h is a t t e n t i o n a l o n g t h e t r a i l le a d i n g t o h is f o r m u l a t i o n o f

t h e S t r o n g L a w .

6. Borel's Chapter III: Continued Fractions

6 . 1 . T h e S e t t i n g o f t h e P r o b l e m

I n C h a p t e r I I I, B O R EL r e t u r n s t o c o n t i n u e d f r a c ti o n s , th e s u b j e c t o f h i s e a r li e r

4 5 - p a g e p a p e r o f 19 03 , " C o n t r i b u t i o n fi l ' A n a l y s e A r i t h m 6 t i q u e d u C o n t i n u . "

T h e c o n t i n u e d f r a c t i o n e x p a n s i o n o f an i r r a t io n a l n u m b e r x i n [ 0, 1 ],

1x - -

a l + l

a 2 + l

a 3 + l

w h e r e e a c h e l e m e n t a , = a , ( x ) is a p o s i t iv e i n t e g e r , is in s o m e w a y s p a r a l l e l t o t h a t o f

t h e d e c i m a l ( o r q - a ry ) e x p a n s i o n . B O R E L c o n s i d e r e d i t as a n i n s t a n c e o f a s e q u e n c e

o f in f i n it e ly m a n y t ri a ls ( o n e f o r e a c h i n t e g e r a n) e a c h o f w h i c h m a y h a v e i n f in i t el y

m a n y o u t c o m e s , e .g . G m a y e q u a l 1, 2 , 3 , . . . , k . . . .

T h e p r o b a b i l i t i e s P~,k, th a t a~ = k , sa t i s fy

~ pi ,k = l .k = l

O n e c o u l d , a pr ior i , m a k e a r b i t r a r y h y p o t h e s e s i n a d d i t i o n , b u t B O R E L a s s e r t e d ,

. .. ; w e a r e g o i n g t o s t u d y t h e h y p o t h e s e s t o w h i c h o n e i s l e d w h e n t a k i n g t h e

g e o m e t r i c p o i n t o f v i e w a l re a d y i n d i c a t e d /t p r o p o s t h e d e c i m a l n u m b e r s .

BOREL (1909: 264) .

T h i s m e a n s , i n e f f e ct , t h a t t h e p r o b a b i l i t y f o r x t o l ie in a n y s u b - i n t e r v a l o f [ 0, 1 ]

is th e l e n g t h o f t h a t s u b - i n t er v a l. 1 O n c e a g a i n t h e m o t i v a t i o n f o r th e e x a m p l e c o m e s

f r o m p r o b l e m s o f " g e o m e t r i c " o r " c o n t i n u o u s " p r o b a b i l i t y , in t h e m o s t n a i v e se ns e.

1 B y e x t e n s i o n , i t a l s o m e a n s t h a t t h e p r o b a b i l i t y t h a t x l i e s i n a B O R E L s e t o f [ 0 , 1 ] i s t h e

m e a s u r e o f t h a t s e t , b u t t h i s a ll i m p o r t a n t a n d n o n - n a i v e e x t e n s i o n o f g e o m e t r i c p r o b a b i l it y b y

m e a n s o f m e a s u r e t h e o r y i s n o t e m p l o y e d o r e x p li c it ly a c k n o w l e d g e d i n t h e p a p e r o f 19 09 . S e e

F A BE R 'S r e m a r k (§ 9 .2 ), m a d e a f t e r r e a d i n g t h e p a p e r o f 1 9 09 , w h i c h e x p l i c i t ly q u e s t i o n s w h e t h e r s u c h

a n e x t e n s i o n c a n b e m a d e o f " d e n u m e r a b l e p r o b a bi l it y " .

I t i s r e a d i l y s h o w n t h a t a s a c o n s e q u e n c e o f t h is a s s u m p t i o n

P l , k = p r o b a b i l i t y { a 1 x ) = k }

= p r o b a b i l i t y < x < ~ - k ( k + 1)"

T h e s e t o f al l x s u c h t h a t { a 2 ( x ) = k} i s a u n i o n o f d i s j o i n t i n t e r v a l s , n a m e l y

~ { a l ( x ) = m , a 2 ( x ) = k } .r a = l

T h e c o r r e s p o n d i n g l e n g t h i s a c u m b e r s o m e i n f i n i t e s e r i e s w h i c h e q u a l s P a , k .

S i m i l a r l y P3,k is g i v e n b y a d o u b l e s e r ie s , c o r r e s p o n d i n g t o t h e s u m o f l e n g t h s o f t h e

d i s j o i n t u n i o n

@ { a l ( x ) = m l , a 2 ( x ) = m 2 , a 3 ( x ) = k } .1 <~ml, m 2 < oo

I n s u m m a r y , t h e p r o b a b i l i t i e s P, , k b e c o m e i n cr e as in g ly u n m a n a g e a b l e w i th

i n c r e a s i n g n , a n d a t t e n t i o n m u s t b e d i r e c t e d t o th e m u l t i - i n d e x e d p r o b a b i l i t y o f th e

s e t o f x s a t i s f y i n g

{ a 1 x ) = m 1 , a 2 ( x ) = m 2 , . . . , a j _ 1 = m j _ 1 , a j ( x ) = k } .

T h e p o i n t s x f o r w h i c h t h e s e s p e c if i ed v a l u e s o f a l ( x ) , . . . , a j ( x ) a r e a c h i e v e d

c o n s t i t u t e a n i n t e r v a l , w h i c h w e s h a l l d e n o t e

[ a 1 = m 1, a 2 = m 2 . . . . a j _ 1 = m j _ 1 , a1 = k J .

D i s t i n c t s u c h i n t e r v a l s w i t h t h e s a m e v a l u e o f j a r e d i s jo i n t, w h i l e [ a 1 = m a . . . . . a j

= m j ] i s a s u b - i n t e r v a l o f [-a 1 = m l , . . ., a j _ a = m ~ _ 1] a n d i n f a c t

[ a l = m 1 . . . , a j 1 = m / _ 1 ] = ( ~ [ a l = m l , . . . , a j _ l = m j _ _ l , a i = m j J .m j~ 1

I n t h i s n o t a t i o n

Pn, k = ~ l [ a l = m l , . . . , a , - l = m , - 1 , a , = k ]m t , . . . , m n - 1

t h e s u m m a t i o n b e i n g o v e r a l l p o s i ti v e in t e g e r v a lu e s o f m l , . . . , m n_ 1 a n d l ( J )

= le n g t h o f th e i n t e r v a l J .

6 .2 . T h e D e r i v a t i o n o f th e K e y I n e q u a l i t y

W h i l e P, , k c a n n o t b e p r e c is e l y c a l c u l a t e d , it c a n b e e s t i m a t e d b y i n t r o d u c i n g t h e

" a p p r o x i m a n t s " o f a c o n ti n u e d f r a c ti o n :

Q , a 1 + 1

a 2 + l

H e r e P n = P n ( a l . . . . . a n) a n d Q , = Q n ( a i . . . . , an ) s a ti sf y t h e c l as s ic a l r e c u r s i o nf o r m u l a e

P , = P , - 2 + a , P n -i , P i = 1 , P o = O ,

Q , = Q n _ 2 + a n Q n j , Q l = a a , Q o = l .

T h e i n t e r v a l [-al = m > . . . , a n _ 1 = m , i , an = k ] h a s e n d - p o i n t s

Pn(mi . . . . , m n _ i , k ) Pn_2-JckPn_l

Q n ( m l . . . . . m n 1 , k ) - Q n _ 2 q - k Q n _ 1

P ~ ( m ~ , . . . , m , _ , , k + l ) P,, 2 + ( k + 1 ) P , _ lQ n ( m ~ , . . . , m , _ i , k + l ) Q , 2 + ( k + l ) Q , _ ~

I n v i r t u e o f t h e c l a s s ic a l i d e n t i t y

1 ( 2 n_ ~ = ( -

t h e l e n g t h o f t h i s i n t e r v a l i s g i v e n b y

1 1. E a l = . l = . l an = k -Q l. . .

k + k + l + Q '°2 ]Q._~!

H e r e ( 2 , _ i = Q , _ l ( m a , m 2 , . . . , m , _ O , Q n _ 2 -- _Q n _ 2 (m l ,m 2 . . . . . rnn 2) a r e con -

v e n i e n t b u t d a n g e r o u s a b b r e v i a ti o n s .

I t f o ll o w s t h a t t h e f a c t o r 1 / Q 2 i is c o m m o n t o b o t h l [ a 1 = m ~ . . . . , a , ~ = m , _ 1,

a , = k ] a n d l [ a z = m i , . . . , a , _ , = m , ,, a , = k + l ] s o t h a t t h e ir r a t io is

S i n c e

k-~Q n 2 ( m l , ' ' ' , m n - 2)

k + 2 + (2 , 2 ( m > . . . ,m , _ 2 )"( 2 , _ l( m j , . . . , m , _ ~)

0 < ~ < 1

f o r a l l c h o i c e s o f m l , . . . , m , _ i , t h i s r a t i o l ie s b e t w e e n

k k+land

k + 2 k + 3

Axio matic Probability 151

f o r a l l c h o i c e s o f m ~ , . . . , m , _ 1 - I t f o l l o w s ~ t h a t

k + lk <Pn,k+l < _ _ ( 6 . 1 )

k + 2 P,,k k + 3

f r o m w h i c h i n e q u a l i t y B O R E L p r o c e e d s . T h e s e t o f n u m b e r s x s a t i s fy i n g G ( x ) = k is a

u n i o n o f i n t e r v a l s a n d n o t i t s el f a n i n t e r v a l ; t h e b a s i c i n t e r v a l s, w h o s e l e n g t h s a r e

c a l c u l a b l e i n t e r m s o f P , ' s a n d Q , ' s , a r e o f t h e f o r m [ a 1 = r e x, . . . , a , 1 = m , _ 1, a , = k -l.

S i n c e e a c h G ( x ) i s t h o u g h t o f a s a " t r i a l " d e t e r m i n e d b y x, s u c h a b a s i c i n t e r v a l

r e p r e s e n t s t h e s e t o f " t r i a l " s e q u e n c e s w i t h p r e s c r i b e d o u t c o m e s o n t h e f i rs t n t r ia l s.

I n th e l a n g u a g e o f C a r t e s i a n p r o d u c t s s u c h s et s h a v e c o m e t o b e c a l l e d cyl inder sets,

w i t h " b a s e " i n t h e p r o d u c t s p a c e o f t h e f ir s t n f a c t o r s p a c e s.

W h a t B O R E L h as c a l c u l a t e d is, f ir st , i n e q u a l i t i e s f o r t he r a t i o s o f p r o b a b i l i t i e s o f

c y l i n d e r s e t s

k l [ a l = m l , . . . , a , _ l = m , _ t , G = k + l ] k + l< . . . . (6.2)

k + 2 l [a~ = m,, 7 .77S 72 mZ ~,~TG~-- -I- < ~ "

T h e c o m p a r a b l e i n e q u a l i t y fo r t h e r a t i o P,,k+~ o f a priori p r o b a b i l i t i e s i s

k + l<Pn, k+l <_

k + 2 P n , k k + 3 "

T h i s l a s t i n e q u a l i t y y i e l d s , b y r e c u r r e n c e ,

( k - 1 ) ( k - 2 ) . . . ( 2 )( 1) < Pn, k (k ) (k - 1) . . . (3 ) (2 )

( k + l ~ . ( ~ ) ) Pn, 1 ( k + 2 ) ( k + l ) . . . ( 5 ) ( 4 )

2 P n k < 6

( k ) ( k + l ) p ,, ~ ( k + l ) ( k + 2 )

1 To state this mor e explicitly: what is established in the text (in not atio n tha t reg rettably suppresses

the indices m l, rn2, ..., m,, 1) is that

k Pr[al=ml ,a~=m 2 . . .. a ,_l= m, ,_l ,G =k+ l ] k + l- - <k + 2 Pr[a l= ml ,aa= m 2 . .. . G_l= m ,_ i , a ,= k] < ~ "

Here the strength of the inequalities lies in the fact that the given bounds hold for all choices of

ms, mz, ..., m,_ ~. It follows th at if one m ultiplies this inequa lity by Pr [a 1 = ml, ..., a,._ 1 = r G - 1, an = k]

and then sums over all possible values of ml, ..., rG_ 1, one obtains

k k + lk+ 2 P"'k<P"'k+l < ~ 3 Pn, k

which is the d esired sta teme nt (6.1). This follow s since

~ - . . ~ P r [ a a = m 1 . . . . a,, l=rn,_ l,a,=k~= Pr[a,= k~.mi - -1 m2= l mn- I

(It should be emphas ized again th at th e set of x for which [ a, = k] is a sum o f disjointopen intervals and

not itself an interval.)

O n e c a n f r e e t h i s o f p , , a b y u s i n g ~ , P , , k = 1. S u m m i n g t h e a b o v e i n e q u a l i t ie sk = l

w e o b t a i n

i .e .

• Pn, k6<k=~ <

( k ) ( k + 1 ) P n,1 ( k + 1 ) ( k + 2 ) '

12 < < 3 ,

P n , 1

½ < Pn , l < ½ •

T h u s i n e q u a l i t i e s a r e o b t a i n e d f o r p . , k :

2 /3 3

( k ) ( k + 1) < P . , k < ( k + 1 )( k + 2 )"

L et P. , k = P. , 1 + Pn, Z+ " " + P .,k = P {a . (x ) < k} . Th en

1 - Pn, k = P ro ba bi l i ty {an(x > k } = P n, k + ~ + P . , k + 2 + ' " "

T h e i n e q u a l it i e s g o v e r n i n g Pn, e i m p l y t h e i n e q u a l i t y

2 __ ~ 2 ~ 3 3

3 ( k + l ) j = k + l 3 (j )( j+ l~ < I - P ~ k < j = k + a ( j + l ) ( j + 2 ) k + 2 "

A t t h i s p o i n t B O R E L i d e n t if i e s e a c h c o n t i n u e d f r a c t i o n w i t h a n i n f i n it e s e q u e n c e

o f d i c h o t o m o u s t r i a ls s o t h a t h e c a n a p p l y h i s A ~ r e su l t. T o d o t h i s h e i n t ro d u c e s a n

i n t e g e r v a l u e d f u n c t i o n ~ b(n ), a n d c o n s i d e r s f o r e a c h f i x e d i r r a t i o n a l x i n t h e u n i t

i n t e r v a l t h e s e q u e n c e o f e v en t s d e f i n e d b y a n ( x ) > qS(n ) ( " succe ss" ) , n = 1 , 2 , 3 . . . .

I f ~b n) i s chosen so tha t

F~ ~(~)

c o n v e r g e s , t h e n

(1 - p . , ~ ( . ) )

c o nv e rg e s. I f ~ n ) d i v e r g e s , t h e n . = 1 ( 1 - P . ,o ( . )) d i v e rg e s . F r o m t h e r e su l t o n A oo

B O R E L c o n c l u d e s

T h e B o r e l C o n t i nu e d F r a c t i o n T h e o r e m . Prob {aN(x > qS(n) i n f i n i t e l y o f t e n } i s

10 o r 1 a c c o r d i n g a s ~ ~ ( n ) c o n v e r g e s o r d iv e r g e s .

BOREL next observes th a t th is sharp a l te rna t ive map be pu t in to even more

1 1str iking form. Indee d, i f ~O-~n) converges, then there exists a O(n) such tha t ~ '

1converges while l i ra ~(n) = 0 an d correspo nding ly, i f ~ ~ diverges, there exists

° ~ ~(n) '

1 ~(n)a ~(n) suc h th at ~ ~ dive rges wh ile ...vlim~(n) - - ~" I t fol lows from the o r iginal

formula t ion appl ied to ~ (n) tha t the theorem can be re formula ted thus :

f an 0 ) Prob~limm a, = o o ; = 1 accordingly as ~ 1P r o b l l i m ~(n ) = ~ = 1 or [ O(n) J ~(n )

converges or diverges.

BOREL wrote o f th is theorem: " In th is form i t appears to me to be the m ost

interest ing o f those we have ob taine d in this mem oir ." BOREL (1909: 269). As a

his tor ica l note BOREL adds " the n ota t i on l im, d ue to Pr ingshe im, denotes ' l a p lus

grande l imi te ' de f ined by Cauchy, and made prec ise by du Bois Reymond and

H a d a m a r d . "

6.3. Defect in Reasoning

The defect in reaso ning h ere is l ike that of the case of the decim al f ract ion, but

more grave . In the dec imal case only the ze ro par t of the Zero-O ne Law was used ,09

wh en th e co nclus ion Ao~ = 0 was asser ted for a sequence of t r ia ls with ~P n < oo.1

Th ou gh the tr ia ls were no t in depe nden t, the "CANTELLI" genera l izat ion o f BOREL'Soo

pr oo f tha t Aoo = 0, whe never ~ p, is convergen t , suff ices to v alidate this aspect of the

reasoning. 1

Fo r the case of the cont in ued f rac t ion , where once aga in the " t r ia l s" und er

cons idera t io n a re dependent , the conc lus ion tha t Aoo - -0 ma y aga in be jus t i fied inoo

th is fash ion , i f ~ p , converges. Howev er the com panio n resul t for cont in ued1 co

fraction s, th at A co = 1 if ~, p, diverges, requires a different gen eral izati on of BOREL'S1

or ig ina l d iscussion of the d ivergent case to cope wi th the d ependence of the " t r ia ls"

(that is, the digits a l , a 2 . . . ). Th at the tr ia ls are dep ende nt is c lear bo th

geom etr ical ly and a lgebraical ly. We shall re turn to this point , and give the required

genera l izat ion of the Aoo = 1 result , in § 8.2 below which deals w ith the wo rk of

F. BERNSTEIN.

6.4. A Possible Clue to the Genesis of the Borel Continued Fraction Theorem

Once again, as in the case of the BOREL Strong Law , one can ask w hat led

BOREL to the rema rkab le ( if f lawed) discussion o f con tinu ed fract ions as an exam ple

of h is general theo ry of denu merab le probabi l i ty . As before , one can s imply appea l

to the intu i t ion of a prof ou nd and fert i le inte l lect . In this case, however , an

a l t e r n a t i v e o f c o n s i d e r a b l e w e i g h t m u s t b e c o n s i d e r e d . T h e r e l e v a n t fa c t is t h a t t h e

S w e d i s h m a t h e m a t i c i a n A . W I M A N h a d o b t a i n e d a n a n a l o g o u s r e su l t d a t in g b a c k

t o 1 9 0 0 - 1 9 0 1 w h i c h a s s e r t s z e r o p r o b a b i l i t y f o r a s e t i n th e u n i t i n te r v a l . W IM A N 'S

s e t ( a s u b s e t o f t h e i r r a t i o n a l s i n t h e u n i t i n t e r v a l ) w a s a l s o d e f i n e d b y as o p h i s ti c a te d c o n d i t i o n o n t h e a s y m p t o t i c b e h a v i o r o f t h e t e rm s a n ( x ) o f t h e

e x p a n s i o n o f i ts m e m b e r s i n c o n t i n u e d f r a c t io n s . T h e c a l c u l a ti o n s s h o w i n g i t t o

h a v e z e r o " p r o b a b i l i t y " o v e r l a p p e d s e v e r a l o f B O R EL 'S . W I M A N ' s p r o o f i n v o l v e d

a n e x p l ic i t u se o f B O R EL 's n e w t h e o r y o f m e a s u r e t o e x t e n d t h e s c o p e o f c l as s ic a l

g e o m e t r i c p r o b a b i l i t y a n d r e li e d o n a s c r u p u l o u s u s e o f c o u n t a b l e s u b - a d d i t i v it y o f

s u c h m e a s u r e i n t h e p r o o f (WIMAN ( 1900 ; 1901)). WIMAN h i m s e l f was r e pa i r i ng

s e r i o u s d e f e c t s in t h e p r e v i o u s w o r k o f h i s c o l l e a g u e T . B R O D t~ N (1 90 0), i n a p o l e m i c

t h a t r a g e d b e t w e e n t h e t w o . T h e p r o b l e m t h e y d e b a t e d h a d a r is e n in t h e c o n t e x t o f

C e l e s t i a l M e c h a n i c s w h e r e i t h a d b e e n r a i s e d b y t h e S w e d i s h m a t h e m a t i c i a n -

a s t r o n o m e r H . G Y L DI2 N; a s a re s u l t t h e i r p o l e m i c w a s d i s p u t e d i n j o u r n a l s r e a d b y

N o r d i c m a t h e m a t i c a l a s t r o m e r s b u t w a s h a r d l y f am i li ar t o t h e g e n e r a l E u r o -

p e a n m a t h e m a t i c a l c o m m u n i t y . I n c o n t r a s t t o t h e c o r r e s p o n d i n g c a s e o f V AN

V L EC K 'S w o r k o n d y a d i c d ig it s, t h e r e is n o d o u b t o f BO R EL 'S a c q u a i n t a n c e w i t h

t h i s p r i o r w o r k ; i n 1 9 0 5 BO R E L e x p l i c i t y g a v e W I M A N p r i o r i t y f o r t h e i n -

t r o d u c t i o n o f m e a s u r e i n to g e o m e t r i c p r o b a b i li ty . T h i s a c k n o w l e d g e m e n t , a c c o m -

p a n i e d b y a b i b l i o g r a p h i c r e f e r e n c e t o W I M A N ' s p a p e r s , o c c u r s i n B O R EL (1 90 5),

w h i c h i s d i s c u s s e d b e l o w ( c f § 7 .3 ). Ho w ev e r , BOREL do es no t , i n 1905 , r e f e r t o

t h e c o n t e n t o f W I M A N ' s re s u lt , n o r d o e s h e e v e n m e n t i o n t h e o c c u r r e n c e o f

c o n t i n u e d f r a c t io n s i n t h e f o r m u l a t i o n o f t h e p r o b l e m s o l v e d b y W IM A N . T h e

m u c h m o r e i n f lu e n t i a l p a p e r o f 1 90 9 la c k s a n y r e f e r e n c e to W I M AN 'S w o r k

w h a t s o e v e r , c o n s i s t e n t w i t h it s o m i s s i o n o f a n y i n t e r p r e t a t i o n i n t e r m s o f

m e a s u r e o f t h e r e s u l t a b o u t c o n t i n u e d f r a c t io n s .

B O RE L s aw h i s C o n t i n u e d F r a c t i o n T h e o r e m a s a n e x a m p l e o f a g e n e r al t h e o r y

o f p r o b a b i l i t y o n a b s t r a ct s ets (t h e p r e s u m e d m e a n i n g o f " p o i n t e d e v u e l o g i q ue " ) .

O n t h e e v i d e n c e p r e s e n t e d a b o v e , h e w a s a t b e s t u n c l e a r t h a t h is g e n e r a l t h e o r y w a s ,

i n th i s c a s e, e q u i v a l e n t t o m e a s u r e t h e o r y . I t i s e n t i r e l y p o s s i b le h e t h o u g h t

d e n u m e r a b l e p r o b a b i l i t y a p p l i e d t o c o n t i n u e d f r a c t io n s o f f er e d a d is t in c t , m o r e

" e f f e c t i v e " (i.e., " c o n s t r u c t i v i s t " ) a l t e r n a t i v e . I n a n y c a s e B O R E L u n q u e s t i o n a b l y

k n e w o f a t l e a st o n e p r o b a b i l i s t i c r e s u lt c o n c e r n i n g c o n t i n u e d f r a c ti o n s b e f o r e 1 9 09

a n d t h is c o u l d w e l l h a v e b e e n t h e g e n e s i s o f h is o w n e x a m p l e o f t h e g e n e r a l t h e o r y .

T h e d i s t i n c t i o n b e t w e e n W I M A N a n d V A N VLECK a s p r e c u r s o r s o f B O RE L ism a r k e d i n tw o w a y s : f ir st , w h i le b o t h e m p l o y e d m e a s u r e t h e o r y , WIMAN,u n l i k e

VAN VLECK, e x p l i c it ly c o n s i d e r e d h is t h e o r e m a s a s o l u t i o n t o a p r o b l e m o f

p r o b a b i l i t y ; s e c o n d , w h i le V A N V L E C K 's w o r k m a y h a v e b e e n k n o w n t o BOREL,

WIMAN's u n q u e s t i o n a b l y w a s k n o w n .

6 .5 . T h e C a n t e ll i M o d i f i c a t io n o f t h e B o r e l Z e r o - O n e L a w

T h e B O R EL Z e r o - O n e L a w h a s u n d e r g o n e c o n s i d e ra b l e g e n e r a l i z a ti o n s in c e its

f ir st f o r m u l a t i o n i n 19 09 . I n p a r ti c u l a r , th e " z e r o " c a s e ( A ~ = 0 w h e n ~ p ,

c o n v e r g e s ) h a s s u b s e q u e n t l y b e e n e s t a b l is h e d w i t h n o a p p e a l t o t h e i n d e p e n d e n c e

o f t h e e v e n t s w h o s e p r o b a b i l i t i e s a r e p 1, P 2 , . .- - T h e s e e v e n t s a r e t h e " t r i a l s " o f t h e

o r i g i n a l BO R EL f o r m u l a t i o n . T h i s r e m o v a l o f t h e h y p o t h e s i s o f i n d e p e n d e n c e i n t h e

" z e r o " c a s e is a t t r i b u t e d t o C A N T E L L I ( 1 9 1 7 a , 1 9 1 7 b ), w h o s e w o r k w e s h a l l d i s cu s s

i n P a r t I I . H o w e v e r w e h a v e o c c a s i o n t o r e f e r t o t h e " C A N T E L L I " r e s u lt a n d

" C A N T E L L I -l ik e r e a s o n i n g " i n w h a t f o ll o w s . T h e r e f o r e w e i n t e r p o l a t e a t t h is p o i n t

t h e s t a n d a r d m o d e r n f o r m u l a t io n , t o r e n d e r t h e i m m e d i a t e l y e n s u in g d i s c u ss io n

s e l f - c o n t a i n e d . T h i s f o r m u l a t i o n d i f f e r s s u b s t a n t i a l l y f r o m C A N T E L L I ' s o r i g i n a l

o n e i n t h a t i t d e p e n d s o n a n u n d e r l y i n g a - a d d i t i v e m e a s u r e .

C a n t e l l i L e m m a . L et ( f2 , ~ , P) be a proba bi l i ty space , so that ~ is a a- f ie ld o f se ts

fro m f2, and P is a countably a ddi t ive non-negat ive norm al ized measure de f ined on theel3

s e ts o f ~ . I f E l , E2 , . . . , En, . . . are sets in ~ and ~ P (E ,) converges, then P ( l i m s u p E n )1 n~oo

= 0. T h e p r o o f is g iv e n a f t e r s o m e p r e l i m i n a r y r e m a r k s .

R e m a r k s . 1) l i m sup E , is de f i n ed a s ( ~ U E , and so i s i n ~ . I t i s t he s e t o f a l ln ~ N = I n= N

p o i n t s w h i c h a r e i n i n f in i te l y m a n y o f t h e E . ' s .

2 ) I f E c l i m s u p E . , a n d E i s i n ~ , t h e n P(E)__<P(lim s u p E n ), s o t h a t u n d e r t h en ~ co n ~ o o

h y p o t h e s i s ~ P ( E . ) c o n v e r g e s , o n e c o n c l u d e s P ( E ) = O .n = l

3 ) I f N i s c o m p l e t e w i t h re s p e c t t o P , t h e n E c l i m s u p E . i m p l i e s b y r e m a r k 2 )

t h a t E e ~ a n d f u r t h er t h a t P ( E ) = 0 . ~ o o

4 ) T h e n o t a t i o n E . h e r e is n o t t o b e c o n f u s e d w i th t h e n o t a t i o n i n t r o d u c e d b y u s

i n o u r d i s c u s s i o n o f B O R E L (1 90 9), w h e r e E ~ w a s t h e o c c u r r e n c e o f p r e c i s e l y n

s u c c e s s e s a n d A , = P(E~). I n f a c t , w e h e r e u s e E , t o p l a y t h e r o l e o f t h e e v e n t " s u c c e s s

a t t h e n h t r i a l " i n B O R E L ' S t e r m i n o l o g y . T h e p r o b a b i l i t i e s P(En) o c c u r r in g h e r e

c o r r e s p o n d t o t h e p r o b a b i l i ti e s p . i n t r o d u c e d b y B O R EL , a n d P ( li m s u p E ~)

c o r r e s p o n d s t o B O R E L's A o o. I t is th i s n o t a t i o n w h i c h w i ll b e u s e d h e n c e f o r t h .

P r o o f o f t h e C a n t e ll i L e m m a . T h e s e ts ~ ) E n d e c r e a s e w i th i n c r e a s i n g N , s o t h a tn = N

P Q ~ N E n ) i s a n o n - i n c r e a s i n g s e q u e n c e a n d

P( l i m sup En) = l i m P En ,n ~ o O N ~ o o n

b y t h e c o u n t a b l e a d d i t iv i t y o f P . F u r t h e r

b y t h e s u b - a d d i t i v i t y o f P , a c o n s e q u e n c e o f c o u n t a b l e a d d i t iv i ty . S i n c e ~ P ( E , )oo 1

c o n v e r g e s b y h y p o t h e s i s , ~ P ( E , ) t e n d s t o 0 w i t h i n c r e a s i n g N . T h u sN

P ( l i m s u p E . ) = l i m P E ,, < l i m P(En) = 0t l ~ o O N ~ o o N ~ o o

as de s i r ed .

15 6 J. BARONE &; A. NOVIKOFF

I f w e r e - i n t roduc e t he no t a t i o n P(En)=p, an d A~o=P(l imsupE,) , t he n w ecO

ob tain the CANTELLI mo dif icat i on of BOREL's resul t : ~ Pn conv erges imp l ies A~o

= 0 , w i t hou t a ny a s s um pt i on a s to i nde pe nde nc e .In § 10.2 below, as wel l as in § 7 imm edi ate l y fol low ing we give ins tances of this

reas oni ng by BOREL, FRt~CHET, and HAUSDORFF, a l l of wh om pre ced ed

C A N T E L L I .

7. Borel ' s Earl ier Works

7.1. Introduction

This sec t ion dea l s wi th two ea r l i e r works of BOREL: "Contr ibut ion & l 'analyse

a r i thm 6t ique du co nt in u" (BOREL (1903) ) and " Re m arq ue s sur ce r t a ines ques t ions

de probabili t6" (BOREL (1905)).

Each of these works wi ll be re la ted to the cons ide ra t ions of BOREL'S "L es

p roba b i l i t 6 s d6nombra b l e s e t l e u r s a pp l i c a t i ons a r i t hm6t i que s " i n o rde r t o g i ve

• fur the r ev idence su ppo r t ing the ana lys is of BOREL (1909) presen ted above .

In the discuss ion of BOREL (1903), the focus wi l l be on the inf ini te sub-additivity

of geom et r i ca l vo lum e. In pa r t i cu la r , we sha ll focus on the resu l t tha t , for su i tab le

sets E, El , E2 . . . . in Eu cl id ean n-d ime nsio nal space, the assum ption s ~ vol(E~) < 0%1

and E c l i ra sup Ei , imp ly v ol (E )= 0. (This i s the resu l t of 1909, nam ely, A~ = 0 in

the case of converg ence , except in geo met r i c an d n ot p robabi l i s t i c t e rms .) The

a bs e nc e o f a ny a s s umpt i ons c o r r e s pon d i ng t o i nde pe nd e nc e i s no t e w or t hy , i.e. this

beco mes a "CANTELLI" resu l t when p ut in a probabi l i s t i c se t ting . I t is prove d b y

BOREL, us ing "CANTELLI- l ike" reason ing a nd some geom et r i c hypoth eses on the

se ts E , E 1, E2 , .. . suf fi c ien t to ensure tha t they h ave "v ol um e" in an e lem enta ry

sense.

I t wi ll be show n, by c i ta t io ns fro m BOREL (1903), that cou nta ble su b-add i t ivi ty,

the ke y to BOREL's pr oo f of the abov e resu l t, was in t imate ly conn ec ted in BOREL's

mind wi th the HEINE-BOREL Th eore m. On th i s ev idence it s eems a t l eas t h ighly

plaus ib le tha t BOREL's re luc tance to employ countable addi t iv i ty and sub-a dd i t i v it y i n de num e ra b l e p roba b i l i t y w a s be c a us e o f t he s t r a nge ne w c on t e x t i n

which the HEINE-BOREL Th eo rem was inappl i cable . Because cou ntab le add i t iv i ty

a nd s ub -a dd i t iv i t y ha d no t p l a ye d a r o l e i n e i the r f in i te o r c on t i nuous p roba b i l i t y

by 1909 1, there was no s t rongly sugges t ive evidence that e i ther was an essent ia l

p r o p e r t y t o d e m a n d o f " d e n u m e r a b l e p r o b a b i l it y " . I n d e pe n d e n c e, o n t h e o t h e r

hand, was a lmos t the ch arac te r i s t i c fea ture of probab i l i ty , and the ex tens ion of

i nde pe nde n c e t o t he de num e ra b l e c a s e e xe r te d a c o r r e s pond i ng l y s tr onge r a ppe a l

to BOREL as the essent ia l ingredient for his new theory in 1909.

In s umm ary , the conten t s of BOREL (1903) he lp con s ide rably in acc oun t ing for

BOREL's t imidi ty about countable sub-addi t iv i ty ( in 1909) , and in provid ingreasons for h i s needles s ly res t r i c t ed as se r t ion about A~ in the convergent case .

* The e xcept ion a l d iscu ss ion of con t in uou s probab i l i ty by WIMAN (1900; 1901) re fe rred to abov e

wil l be t r ea ted in a sepa ra te note .

Ax ioma tic Prob ability 157

T h e p a p e r B O R E L ( 19 05 ) r e p r e s e n t s B O R EL 's c h i e f c o n t r i b u t i o n t o p r o b a b i l i t y

p r i o r t o 1 90 9. T h i s p a p e r r e p r e s e n t e d ( fo r al l i ts s h o r t c o m i n g s ) a s i g n i f i c a n t a d v a n c e

i n t h e f o u n d a t i o n s o f p r o b a b i l i t y t h e o r y w h i c h h a s b e e n u n a c c o u n t a b l y n e g l e c te d .

I t p r o p o s e d t h a t i n t h e u n i t i n t e rv a l t h e m e a n i n g o f (g e o m e tr ic ) p r o b a b i l it y ,i d e n t i f i e d w i t h l e n g t h , b e s u b s t a n t i a l l y g e n e r a l i z e d t o b e i d e n t i f i e d w i t h h i s n e w

t h e o r y o f m e a s u re .

B O R E L (1 90 5) u n d e r s c o r e s h o w s i g n i f i c a n t w a s t h e g a p b e t w e e n B O R E L's v ie w o f

( g e o m e t r i c ) p r o b a b i l i t y o n t h e u n i t i n t e r v a l , o n t h e o n e h a n d , a n d h i s v i e w o f

d e n u m e r a b l e p r o b a b i l i t y o n t h e o t h e r . W i t h B O R E L (1 90 5) i n m i n d o n e c a n b e

a l m o s t c e r t a i n t h a t B O R E L 'S r e s e r v a t i o n s ( i n 1 9 0 9 ) a b o u t t h e r o l e o f c o u n t a b l e

a d d i t i v i ty i n " d e n u m e r a b l e p r o b a b i l i t y " s t em f r o m a n i n c o m p l e t e r e c o g n i ti o n t h a t

t h e m a c h i n e r y o f t h e " g e o m e t r i c p o i n t o f v i e w " , a p p li c a b le i n h i s d e c i m a l a n d

c o n t i n u e d f r a c t i o n e x a m p l e s , s h o u l d b e a v a i l a b l e a s w e l l i n t h e g e n e r a l t h e o r y .

T h e a i m o f t h e d e t a i l e d d i s c u s s i o n o f t h e s e e a r l ie r w o r k s i s t h u s t o s h e d l i g h t o n

t h e s h o r t c o m i n g s o f B O R E L (1 90 9) ( n o t e d a b o v e ) b y c o m p a r i n g t h e m w i t h

v i e w p o i n t s B O R E L h i m s e l f p o s s e s s e d b e f o r e 1 90 9.

7 .2 . Bore l (19 03 ) : F oreshadowings o f Can te l l i-L i ke Reason ing

I n B O R E L (1 90 3), c o n t i n u e d f r a c t i o n s h a d b e e n t h e m a i n o b j e c t o f s t u d y a n d n o t

m e r e l y a s o u r c e , a m o n g o t h e r s , o f e x a m p l e s a s t h e y w e r e i n B O R E L ( 19 09 ). T h e

g e n e r a l p u r p o s e o f B O RE L (1 90 3) w a s t o e s t a b l i s h t h e e x i s te n c e o f c o v e r i n g s o f t h e

i n t e r v a l [ 0 , 1 ] b y s u b - in t e r v a l s , m e m b e r s h i p i n w h i c h d e m a n d s a h i g h d e g r e e o f

a p p r o x i m a b i l i t y b y r a t i o n a l s . A t y p i c a l s u c h r e s u l t is , e.g., t h a t every r e a l n u m b e r c~

p o s s e s s e s r a t i o n a l a p p r o x i m a n t s P /Q s u c h t h a t

- ~ < 1 /~ Q 2

wh ere (2 can be req u i red to l ie in a p r e sc r ibe d in t e rv a l (A , B ) , sa t i s fy ing

I O < A < 1 5 A 2 < B .

I n c o n s e q u e n c e , o n e c a n p r e s c r i b e a n i n f i n it e s e q u e n c e o f in t e r v a l s ( A , , B n),s a t i s f y i n g t h e a b o v e , a n d t h e r e w i l l e x i st a s e q u e n c e o f c o r r e s p o n d i n g a p p r o x i m a n t s

T h e m a i n t o o l u s e d i n B O R EL ( 19 03 ) is t h e r e p e a t e d u s e o f f i n it e o r d e n u m a r a b l e

c o v e r s o f [-0 , 1 ] ( o r i ts n - d i m e n s i o n a l a n a l o g , t h e n - d i m e n s i o n a l c u b e) . T h e s e c o v e r s

a r e u s e d t o a s s e r t r e l a t i o n s b e t w e e n t h e l e n g t h s o f t h e c o v e r i n g i n t e r v a l s a n d t h e

l e n g t h ( o r , i n h i g h e r d i m e n s i o n s , v o l u m e ) o f s e ts w h i c h a r e e i t h e r c o n t a i n e d i n a

f in i t e subco ve r , o r , a l t e rn a t ive ly , cove re d in f in i t e ly o f t en .

T h e s e n o t i o n s h a d a l r e a d y b e e n e x p l o it e d b y B O R E L i n b o t h h i s c e l e b r a te d

thesis (BOREL (1895)) an d in h is Lemons sur la Th~orie des Fonct ions (BOREL (1898)).

W h a t i s n o w c a ll e d th e H E IN E -B O RE L T h e o r e m h a d a l r e a d y b e e n s t a t e d b y B O R E L,

f o r d i m e n s i o n n = 1, i n h i s t h es i s a n d i n B O R E L ( 18 98 ). T h e r e is n o q u e s t i o n t h a t t h e

1 5 8 J . B A R O N E & A . N O V I K O F F

H E I N E -B O R E L T h e o r e m ( in o n e d i m e n s i o n ) w a s c e n t r a l t o B O R EL 's d e f i n i ti o n o f

m e a s u r e , a n d t o t h e d e f i n i ti o n o f t h o s e s e ts ( s in c e c a ll e d " B O R E L - m e a s u r a b l e " ) t o

w h i c h h e a p p l i e d t h i s t h e o r y o f m e a s u r e . A l t h o u g h B O R EL f a il e d t o g e n e r a l i z e hi s

t h e o r y o f m e a s u r e t o h i g h e r d i m e n s i o n s , h e d i d g e n e r a l i ze t h e H E I N E -B O R E LT h e o r e m , a n d f r o m i t h e d e d u c e d a f o r m o f c o u n t a b l e s u b - a d d i ti v i ty i n h ig h e r

d i m e n s i o n s . T h e e x t e n s i o n o f t h e H E I N E -B O R E L T h e o r e m ( u si n g o n l y o p e n c o v e r s

o f a n e s p e c i a l l y s i m p l e s o r t ) o c c u r r e d i n B O R E L (1 90 3). T h e r e h e g a v e t h e e x t e n s i o n

( T h e o r e m V I I I ) t o b o u n d e d c l o s e d s et s i n d i m e n s i o n n , r e s tr i c ti n g h i m s e l f t o

d e n u m e r a b l e c o v e r s. (B O R EL l a t e r a c k n o w l e d g e d L E BE SG U E 's g e n e r a l i z a t i o n o f t h e

r e su l t to a p p l y t o n o n - d e n u m e r a b l e c o v e r s as w e ll .)

O f s p ec i a l i n t e r e s t is t h e o c c u r r e n c e o f t h e t h e o r e m o n v o l u m e s s t a t e d a b o v e .

T h i s i s T h e o r e m X I b is i n t h e n o t a t i o n o f B O R E L (1 90 3). A s r e m a r k e d i n th e

i n t r o d u c t i o n , i t is a s p e c ia l c a s e o f w h a t is n o w c o n s i d e r e d t h e " C A N T E L L I " p a r t o f

the BOREL-CANTELLIL e m m a s .

T h e s p e c i a li ti e s t h a t s u r r o u n d t h is t h e o r e m o f BO R EL l ie in t h e a s s u m p t i o n s ,

d e r i v i n g f r o m t h e c o n t e x t o f t h e p a p e r o f 19 03 a s a w h o l e , t h a t E l , E 2 . . . . . En, . . . a r e

" d o m a i n s " i n s o m e f ix e d E u c l i d e a n s p a ce s, R k, o f d i m e n s i o n k . A " d o m a i n " is

d e f i n e d as a c l o s e d b o u n d e d p a r t o f k - s p a c e d e t e r m i n e d b y a f in i te n u m b e r o f

a l g e b r a i c i n e q u a l i t i e s

qSj(x 1, . - . , xk) > 0 j = 1, 2 . . . . M .

T h e v o l u m e o f a d o m a i n is a n n - d i m e n s i o n a l ( RIE M A N N) i n t e g ra l , w i th

i n t e g r a n d 1, o v e r th e i n t e r i o r o f t h is d o m a i n . B O R EL c o n s i d e r e d o n l y d o m a i n sw h i c h , i n m o d e r n t e r m i n o l o g y , h a v e n o n - e m p t y i n t e r i o r s a n d w h i c h a r e t h e

c l o s u r e s o f t h e i r i n t er i o r s. I n p a r t i c u l a r , e v e r y d o m a i n c o n s i d e r e d h a s a s t ri c tl y

p o s i t i v e v o l u m e . ( T h u s , f o r e x a m p l e , c l o s e d s p h e r e s a n d c l o s e d c u b e s a r eco

" d o m a i n s " . ) B O R E L s h o w e d t h a t i f ~ v o l ( E , ) is c o n v e r g e n t , t h e n l im s u p E ~ = H1 8 4 o o

h a s " s m a l l c o v e r i n g " . S p e c if ic a ll y , g i v e n e > 0 , t h e r e a r e d o m a i n s

H 1, H a . . . . . H . . . . . s u c h t h a t

H c U ( in t er io r H n), a n d ~ v o l ( H , ) < e .

T h i s m a y b e c a l l e d B O R E L 'S v e r s i o n o f t h e C A N T E L L I L e m m a . B O R E L's o r i g i n a l

f o r m u l a t i o n w i l l b e p r e s e n t e d b e l o w .

W e n o t e t h a t B O RE L s t o p p e d s h o r t o f c o n c l u d i n g v o l ( H ) = 0 . I n d e e d h e c o u l d

n o t d o t h i s si n ce th e p o i n t s e t H n e e d n o t b e a d o m a i n , a n d s o v o l ( H ) n e e d n o t e x i s t

a s a R IE M A N N i n t e g ra l . A t t h e d a t e o f t h is p a p e r t h e t h e o r y o f m e a s u r e h a d n o t b e e n

g e n e r a l i z e d t o n - d i m e n s i o n a l E u c l i d e a n s p a c e, n o r h a d t h e th e o r y o f t h e

L E B ES G U E i n t e g r a l b e e n e s t a b l i sh e d .

A p a r t f r o m t h e re s t ri c ti o n t o " d o m a i n s " t h e t h e o r e m is t h u s v e r y m u c h t h e

C A N TE LL I L e m m a . W h a t is e m i n e n t l y r e m a r k a b l e is t h e l in e o f r e a s o n i n g

e m p l o y e d b y B O R EL . T h e " C A N T E L L I " p r o o f g i v e n in § 6.5 a b o v e , w h e n s p e c i a li z e dt o t h e c a s e a t h a n d , p r o v i d e s t h e d e s i re d s e t o f d o m a i n s H 1 H 2 , . . . , H . . . . . a s

H i = E N + i w h e r e N i s c h o s e n so t h a t ~ v o l ( E N + i ) < e .i = 1

Axi om a t i c P roba b i li t y 159

T h e i n t e re s t i n g q u e s t i o n i s h o w B O R E L a r r a n g e d t o p r o v e t h i s t h e o r e m . T o

e x a m i n e t h i s , i t i s n e c e s s a r y t o r e a d h i s e x a c t f o r m u l a t i o n o f t h i s r e s u l t a n d i ts

i m m e d i a t e p r e c u r s o r , T h e o r e m X I .

T h e o r e m X I . L e t E b e a d o m a i n a n d E l , E 2 , . . . , E h . . . . d o m a i n s s u c h t h a t e v e r y

p o i n t o f E i s i n t e r io r t o a n i n f i n i t y o f t h e m ; t h e n o n e c a n a s s e r t th a t t h e v o l u m e s

v ~ , v 2 , . . . , V h, . . . o f t h e s e d o m a i n s a r e s u c h t h a t t h e s e r i e s

V JC- V2 ~ - . . . -Jr Vh ~- . . •

i s d i v e r g e n t .

T h e o r e m X I bis . L e t E l , E 2 , . . . , E h . . . . . d o m a i n s w i t h v o l u m e s V l , v 2 , . . . , V h, . ..

b e s u c h t h a t t h e s e r i e s

V -}- V 2 -Jf- . . . ~- Uh @ . . .

i s c o n v e r g e n t ; t h e n o n e c a n a s s e r t t h a t t h e s e t H o f p o i n t s w h i c h b e l o n g t o a n

i n f i n i t y o f t h e s e d o m a i n s i s s u c h t h a t , b e i n g g i v e n e a r b i t r a r i l y s m a l l , o n e c a n

c o n s t r u c t d o m a i n s H a , H 2 . . . . . H . . . . . . f i n i t e o r d e n u m e r a b l y i n f i n i t e in n u m b e r ,

s u c h t h a t e v e r y p o i n t o f H i s i n t e r io r t o o n e o f t h e m a n d t h a t , in a d d i t i o n , V~ b e i n g

t h e vo l u m e o f H ~ o n e h a s

v l + v 2 + . . . + v ~ + .- . < e .

( I t a l i c s i n t h e o r i g i n a l . ) B O R E L ( 1 9 0 3 : 3 6 2 ) .

T h e f u l l l in e o f d e v e l o p m e n t l e a d i n g t o th e s e t h e o r e m s p r o v i d e u n a s s a i l a b l e

e v i d e n c e a s to h o w B O R E L, i n t h i s c o n t e x t , a s s o c i a t e d s u b - a d d i t i v i t y a n d t h e

" C A N T E L L I L e m m a " w i t h t h e H E I N E - B O R E L r e su l t . T h i s l in e o f d e v e l o p m e n t i s t h e

f o l l o w i n g c h a i n o f t h e o r e m s , o c c u r r i n g i n S e c t i o n 1 9 o f B O R E L (1 9 03 ).

T h e o r e m V I I I . L e t E b e a g i v e n c l o s e d b o u n d e d s e t, a n d E l , E 2 , . . . , E p . . . . a

d e n u m e r a b l e i n f i n i t y [ f o o t n o t e o m i t t e d ] o f s e t s s u c h t h a t e v e r y p o i n t o f E i s

I N T E R I O R t o a t l e a s t o n e o f t h e m ; i t i s p o s s i b l e t o f i n d a m o n g E l , E 2 , . . . , E p , . . .

a F I N I T E n u m b e r o f s e t s s u c h t h a t e v e r y p o i n t o f E is i n t e r i o r t o a t l e a s t o n e o f

t h e m . ( I t a l ic s i n t h e o r i g i n a l . ) B O R E L ( 1 9 0 3 : 3 5 7 ) 1

This is as c lear a rende ring o f the n-d imen sional HEINE-BOREL heo rem as could b e wished. BOREL

even supplemented i t wi th an extens ion to se t s which a re only c losed (or bounded) a f te r a su i table

projec t ive t ransformat ion . ( In mode rn terms, he cons ide red projective 2-space, or more generally, n-

space.) Such sets he cal led projectively closed o r projectively bounded.

He then g ave a s imple example of three se ts in the pro jec t ive p lane w hich a re projectively closed and

projectively bounded and col lec t ive ly cover the p lane ( two d imensions b e ing chosen for ease of

expos it ion). Thus he could ap ply The orem VII I to each of these se ts . BOREL hen s ta ted a "gen era l ized"HEINE-BORELt he o re m:

The ore m V I I I b i s. If one has a denumerable inf ini ty of sets El , E 2 , . . . , E n , . . . such that every

point of the plane is in the I N T E R I O R of at least one of them (the points at infinity included, ofcourse), one can determine among the E i a fin ite number o f sets such that every point of the plane

is interior to one of them. (Italics in th e original.) BOREL (1903: 359).

Thus BOREL was aw are tha t the se t t ing of the HEINE-BOREL heorem could b e widened f rom n-

dimensions to a t l eas t ce r ta in a l te rna t ive spaces w i thout ch anging the na ture of the asser tion .

N e x t , r e s t r i c t i ng c ons i de r a t i on t o doma i ns a nd t he i r a s s oc i a t e d vo l ume s ,

BOREL es tabl ished his f i rs t (and key) resul t for sub-addi t ivi ty:

T h e o r e m I X. Wh e n a d o ma i n E i s s u c h t h a t e a c h p o i n t i s i n t e r i o r t o a d o ma i n E i( i = 1 , 2 , 3 . . . . , n . . . ) , one can asser t tha t the s um o f the v o lum es o f the d oma ins E i i s

g r e a t e r t h a n t h e v o l u m e o f E . (Ital ics in the original.) BOREL (1903: 360).

I n o t he r w ords ,

impl ies

E c U ( inter io r(E0)1

vol(E) < y~ vol(E3.1

Th e pro of , to be supp l i ed by the reader , involves the use of the HEINE-BOREL

t h e o r e m ~Th eo re m VII I ) a s a pre l imin ary to as sure the ex i s tence of an N such tha t

E c ~) ( inter io r (El)).1

F rom t h i s one c onc l ude s (p r e s uma b l y by e l e me n t a ry c a l c u l us )

vol(E ) < ~ vol(Ei)1

and the resul t fol lows.

T he s t a t e me n t o f T h e or e m IX is to be i n t e rp r e t e d a s i nc lud i ng t he c a s e

vol(E /) = + oo.1

BOREL then poin ted out tha t one can g ive th i s same asse r t ion an occas iona l ly

more c onve n i e n t f o rm a s fo l l ow s :

T he ore m IX b i s. Gi v e n a d o ma i n E a n d a d e n u me r a b l e i n f i n i t y o f d o ma i n s

E l , E 2 , . . . , E n , . ..

s u c h t h a t o n e h a s

• vol(Ei) <vol(E),1

t h e n t h e r e a r e p o i n t s o f E n o t i n t h e i n t e r i o r o f a n y E i. (Ital ics in the original.)BOREL (1903: 361).

Thi s i s, o f course , no m ore than a cont ra pos i t ive fo rmu la t ion o f the or ig ina l

T he ore m IX , r e qu i r i ng no fu r t he r p roo f .

T h i s pa r t i c u l a r f o rmul a t i on c a n be i mme d i a t e l y s ha rpe ne d by us e o f a n

e lem enta ry a r gum ent w hich em ploys s l igh tly l a rge r doma ins E~ conta in ing El .

T he s ha rpe ne d fo rmul a t i on i s :

T h e o r e m X . I f E has vo lume v, and a denumerab le infinity of domain s E i

( i = 1, 2, . . . , n, . . . ) having volumes vl, are such t h a t

~ v i < v1

the there are points o f E b elonging to none of the E i . (Ital ics in the original.)

BOREL (1 90 3:3 61 362).

Theorem X has a s l igh t ly s t ronger conc lus ion than i t s immedia te predecessor

s ince i t avoids re fe rence to the " in te r ior s " of the se ts E~. Thi s n on- to polo gica lve rs ion i s the one which s t r ikes the contempora ry reader as foreshadowing the

genera l i za t ion to measure theory (or probabi l i ty ) . As we have shown, however , i t

w a s a c h i e ve d on l y a f te r T h e or e m IX bis ( involv ing " in te r iors " ) , and th i s in turn

depended c ruc ia l ly on a HEINE-BOREL argument involv ing "open covers" .

T h e o r e m X I a n d X I bis, c i t ed above , a re now d i rec t consequences of the

preceding. BOREL leaves thei r proofs to the reader .

Summ ar iz ing , BOREL wel l knew tha t in ce r t a in c i rcum s tances i f the se t s E ko0

satisfy ~ m(Ek) < oe, th en th e set H = l im sup Ek, of those p oin t s in inf in i te ly m any of

1the Ek'S, has covers of a rb i t ra r i ly smal l to ta l vo lum e. He d id not s t a t e th i s in the

genera l i ty of measu re theory . Th ere i s no reason to suppo se tha t BOREL so muc h as

c onc e i ve d o f " a bs t r a c t i n g" me a s u re t he o ry t o t he de g re e o f ge nera l it y ne e de d fo r

the CANTELLI imp rov em ent of h i s Zero- On e Law. Indeed , he d id n ot even cons ide r

the s t ra ight forw ard genera l i za t ion of measu re theo ry f rom 1 to n d imens ions . Even

i f he had conce ived of n-d imens ion a l measure , the remain in g s t ep to an ab s t rac t

me asu re re ma ins imm ense. In pa r t icular , the absen ce of a HEINE-BOREL resu l t in

the abs t rac t case might have proved an unbr idgeable gul f , for i t was th i s resu l t

which was bas ic to BOREL's meth ods of proof . The conc lus ion seems inescapable

tha t BOREL knew the "CANTELLI" theo rem in a geom et r i c se t ting only , where

topolo gica l cons ide ra t ions were re levant and ava i l ab le . The l ack of the nota t ion for

l im sup E k for a co l l ec t ion of event s fur the r d i sgui sed the resemblance be tween

the se t H of h i s paper of 1903 and the no wh ere des igna ted se t ( ° 'i n fin it ely ma ny

successes" ) in the paper of 1909 whose probabi l i ty i s A~ .

We a re thus l ed to add an ad di t iona l sp ecula t ion co ncern in g BOREL'S t rea tm ent

of h i s Zero -On e Law : the ab sence of a topolog y in the space of tr i a ls prev ented h im

from es tab l i sh ing a HEINE-BOREL Th eo rem ; th is absence in turn b lock ed the way

for h i s use of coun table sub-ad di t iv i ty in the se t ting of " probab i l i t6 d6n om brab le" ,

e ve n i f he ha d t hough t o f p roba b i l i t y a s a na l ogous t o vo l ume (o r me a s u re ) .

In v iew of the Theorem XI h i s c i t ed above , in which sub -addi t iv i ty i s proved , we

are s im ul tan eou sly led to the surpris ing con clus ion tha t BOREL was as c lose in 1903

to a r igo rous p roo f of the "CANTELLI" ve rs ion of h i s Zero- On e L aw as was

CANTELLI in 1917. The difference is that CANTELLI's work deals explicit ly with

prob abi l i ty a nd so was t aken up by succeeding probabi l i s t s, whi le BOREL'S pap er of

162 J. BARONE 8¢ A. NOVIKOFF

1 9 03 , w i t h i ts i n t ri g u i n g T h e o r e m X I bis , r e m a i n e d r e l a t i v e l y n e g l e c t e d . I ts t it l e a n d

e v e n i t s m a i n r e s u l t s g i v e n o c l u e t o i ts i n t e r e s t i n g i n t e r n a l a r g u m e n t s 1.

7 .3 . Bore l (19 05 ) : A n Ea r ly Iden t i f i ca t ion o f Geometr ic P robab i l i ty w i th M easure

T h e p a p e r o f 1 90 5 c o n t a i n s t h e f o l l o w i n g f o u r a s s e r ti o n s o f i n t e re s t :

1) I f o n e u s e s th e c o n v e n t i o n t h a t t h e p r o b a b i l i t y o f a s e t is p r o p o r t i o n a l t o

i ts l e n g t h ( o r a r e a , o r v o l u m e ) , it s h o u l d b e m a d e e x p l i c i t t h a t t h i s is a

c o n v e n t i o n a n d n o t t h e i n tr in s ic m e a n i n g o f p r o b a b i li ty .

A l l t h e p r e c e d i n g is w e l l - k n o w n , b u t q u e s t i o n s o f p r o b a b i l i t y h a v e g i v e n

r is e to s o m a n y v e r b a l c o n t r o v e r s i e s a r is i n g s i m p l y f r o m a l a c k o f a g r e e m e n t

o n t h e c o n v e n t i o n s o f l a n g ua g e , t h a t i t m a y n o t b e s u p e r f lu o u s t o m a k e

p r e c i s e t h e c o n c e p t s w h i c h I w i ll e m p l o y . B O R E L ( 1 9 0 5 : 1 2 4).

2 ) F o r t h e s e t E o f r a t i o n a l n u m b e r s i n [ 0, 1] , a n d w i t h t h e a b o v e c o n v e n t i o n ,

P(E ) = j f (x) dxo

w h e r e

= t h e ch a r a c t er i st i c f u n c t io n o f E = ~ I i f x is r a t io n a lf ( x )(o i f x i s i r r a t i o na l .

S i m i l a r l y ,1

P ( E c) = ~ F ( X ) d xo

w h e r e

f ( x ) = ~ l i f x is i r ra t i o n a lF ( x ) = l -

1 0 i f x i s r a t i on a l .

T h e s e i n te g r al s p r o d u c e t h e " r d p o n s e e v i d e n t e " t h a t P ( E ) = O, P ( E c)= 1 , b u t n o t i f

o n e r e s t r i c t s o n e s e l f t o R I E M A N N i n t e g r a t i o n .

H o w e v e r , i f o n e u s es t h e n e w d e f i n i t io n o f i n t e g r a l w h i c h i s d u e t o L e b e s g u e ,

o n e s e e s th a t e a c h o f t h e f u n c t i o n s f ( x ) a n d F ( x ) i s i n t e g r a b l e i n t h e s e n s e o f

L e b e s g u e , o r , m o r e b r ie f ly , L - i n t e g r a b l e a n d t h e i r L - i n t e g r a l p r o v i d e s t h e

c o r r e c t [sic] m e a n v a l u e o r p r o b a b i l i ty s o u g h t. L e b e s g u e 's m e t h o d s t h u s a l l o w

u s t o a p p r o a c h q u e s t i o n s a b o u t p r o b a b i l i t y w h i c h a p p e a r i n a c c e s s i b l e t o t h e

c l a s s ic a l p r o c e d u r e s o f i n t e g r a t i o n . M o r e o v e r , i n c e r t a i n o f t h e s i m p l e s t c a s e s, it

s u f fi c es t o u s e t h e t h e o r y o f s e ts t h a t I c a l l e d measurable a n d w h i c h L e b e s g u e h as

n a m e d B-measurable; t h e a p p l i c a t i o n o f m e a s u r a b l e s et s t o p r o b a b i l i t y t h e o r y

w a s f i r st m a d e , t o m y k n o w l e d g e , b y W i m a n 2. B O R E L ( 1 9 0 5 : 1 2 5 - 1 2 6 ).

1 It shoul d be observed th at CANTELLI himse lf generalized the inequalit y

P(U E,) < Z P(Ek)

from finite unio ns to d enumera ble ones in the s ame easygoing argument -by-ana logy fashion as BOREL

generalized finite independence to count able independence. Th at is, with no effort at proof, and

specifically with no reference to measure theory.

2 Aut hor s' footnote: In a footnote BOREL cites WIMAN (1900; 190i). These papers, and mor e

generally, the polemic between WIMAN and his colleague BRODt~N will be discussed in a s eparate

publication.

This shows tha t BOREL regarded the re la t ion

1P(E) = S f~(x) dx

w he re

f~ (x ) = { 10 ififx(~EXSE

as applicable to at least al l BOREL sets.

Inc identa l ly , BOREL regards the de te rmina t ion P(E)=0 for the ra t iona l s a s

c l e a rl y " c o r r e c t " w i th no s us t a in i ng a rgum e n t o t he r t ha n a ppe a l t o t he a u t ho r i t y o f

POINCARI~, wh o had inde ed rega rde d i t as evident , before the existence of a t he o ry

of measure . W e h ave ex am ine d POINCARI~'s lec ture n otes , Calcul des Probabilitds(1893-1894 lectures), a t the Ins t i tu t Henr i Poincar6 , Pa r i s , where th i s en igmat ic

p ron oun c e m e n t i s t o be found , a l r e a dy i nc o rpo ra t e d i n the t e x t a t tha t " p r e m a t u re "

da te , some 16 years be fore the publ i ca t ion of h i s Calcul des Probabilit&

3) An exam ple is considered, n am ely th e se t E ("), def ined for each inte ger valu e

of the p a ram ete r n as a ll rea l numb ers e in [-0 , 1] such tha t the re a re re la t ive ly pr ime

integers p, q satisfying

<1q 1 q"

For each in teger va lue of n , i t i s ev ident tha t

E (") c ~ l~)qw he re

I~½ = p 1 p t- ,q" ' q

p, q are re la t ively pr ime, i.e., (p, q) = 1, an d the u ni on is ov er all such pairs p and q. It

i s then as se r t ed (wi thout comment ) tha t

)=q 2 -P(E(')) < 2 1(I~½ O(q) q.(p , q ) = 1

He re ~b(q) = the n um be r of integers less tha n and re la t ively pr im e to q, and the ser iesclear ly con verg es i f n > 2. This i s a wond erfu l ly c lear ex amp le o f BOREL's expl ic i t

use of coun table sub-addi t iv i ty , bu t only in the contex t of geom et r i c probabi l i ty , o r

wha t he was to ca l l in 1909, the "poin t de rue g6om6t r ique" .

I t i s unf or tun a te tha t BOREL did n ot re l a te th i s examp le in the pap er of 1905 to

his ideas of 1903 discussed abov e, by conc ludin g fur t her th at th e se t of poin ts c~

sat is fying for a ny f ixed n > 2 the ine qua l i ty

fo r infinitely many d i s t inc t ra t iona l s p/q has measure zero. As a resul t , we lack

d o c u m e n t a r y e v id e n ce o f a "CANTELLI" - l i ke asser t ion by BOREL that

P(l im sup E. ) = 0 i f ~ P(E,) < oo

164 J. BARONE8~; A. NOVIKOVV

i n a m e a s u r e - t h e o r e t i c i n t e r p r e t a t i o n o f p r o b a b i l i t y b y B O RE L , e v e n i n t h e c a s e o f

g e o m e t r i c p r o b a b i l i t y w h e n h e w a s e x p l i ci tl y a w a r e o f s u c h o n i n t e r p r e ta t i o n .

4 ) A r e m i n d e r t h a t B - m e a s u r a b l e s e t s f o r m , i n c o n t e m p o r a r y t e r m s , a a -

a l g e b r a , a n d t h a t t h e a s s o c i a t e d m e a s u r e i s , a g a i n i n c o n t e m p o r a r y t e r m s , o r -

a d d i t i v e .

I n s u m : B O R E L 's i d e n t i f i c a t i o n o f g e o m e t r i c p r o b a b i l i t y w i t h m e a s u r e is e x p li c i t

i n 1 90 5, a n d f u r t h e r t h e s c o p e o f (g e o m e t r i c ) p r o b a b i l i t y i s e x p l ic i tl y e n l a r g e d , t o

a p p l y t o a a - a l g e b r a o f s e ts ( c f fo ot no te 1 , § 6 .2) .

O u r a n a l y s is o f t h e p a p e r o f 1 90 9 a b o v e s h o w s b y c o n t r a s t t h a t o n l y i n t h e

s p e ci a l a p p l ic a t i o n s ( n a m e l y d e c i m a l e x p a n s i o n s a n d c o n t i n u e d f r a c ti o n s w h e r e t h e

c o n v e n t i o n i s e x p l ic i t ly m a d e t h a t t h e p r o b a b i l i t y o f a n i n t e r v a l i s i ts l e n g t h ) is it

e v e n m a r g i n a l l y p o s s ib l e t h a t B O R E L h a d t h e s a m e f a c ts i n v ie w . In t h e p a p e r o f

1 9 0 9 t h e s e e x a m p l e s a r e c o n s i d e r e d o n l y a f t e r t h e f u n d a m e n t a l Z e r o - O n e L a wc o n c e r n i n g A ~ i s o b t a i n e d . T h i s r e s u l t is in t u r n c o n s i d e r e d i n d e p e n d e n t l y o f

g e o m e t r i c c o n s i d e r a t i o n s a n d i n t h e s e e m i n g c o n v i c t i o n t h a t c o u n t a b l e i n d e -

p e n d e n c e i s th e k e y n o t i o n , c a s t i n g s u b - a d d i t i v i t y a n d o - - ad d i ti v it y a s i de . F o r

B O RE L , t h e " p o i n t d e v u e l o g i q u e " h a d e c li p se d t h e " p o i n t d e v u e g 6 o m 6 t r i q u e " i n

1 90 9 ev e n t h o u g h b o t h w e r e a p p l i c a b l e a n d i n s p it e o f t h e a d d i t i o n a l i n s ig h t s

( r e c o g n i z e d i n 1 9 0 5 ) o f f e r e d b y t h e l a t te r .

8 . The Bore l -Berns te in Polemic

8 . 1 . I n t r o d u c t i o n

B O R EL 'S r e s u l t s c o n c e r n i n g d e c i m a l ( o r d y a d i c ) d ig i t s a n d c o n t i n u e d f r a c t i o n s

a t t r a c t e d c o n s i d e r a b l e s t i r i n t h e y e a r s i m m e d i a t e l y f o l lo w i n g 1 9 0 9 . I n 1 91 0 FA B E R

p r o v e d a g a i n t h e r e s u l t c o n c e r n i n g t h e d e c i m a l d i g it s, t h o u g h i n a s u b s t a n t i a l l y

d i f f e r e n t w a y ( c f §9 .2 ). In 191 1 F . BERNSTEIN a t t a ck ed BOREL's p ro o f o f the

C o n t i n u e d F r a c t i o n T h e o r e m , s u p p l y i n g a n a l t e r n a t i v e o f h i s o w n . B O R E L

r e s p o n d e d t o B E R N S TE IN 's p a p e r w i t h a m o d i f i c a t i o n o f h is Z e r o - O n e L a w ( 1 91 2),

a d d i n g t h a t t h i s m o d i f i c a t i o n , c o u p l e d w i t h i n e q u a l i t i e s a l r e a d y i n t h e p a p e r o f

1 90 9, s u ff i c ed t o v a l i d a t e h i s r e s u l t o n c o n t i n u e d f r a c t i o n s . T h e e x c h a n g e w i t h

BERNSTEIN of fe rs a n o p p o r t u n i t y t o e x a m i n e o n c e a g a i n B O R E L ' s o w n i n -

t e r p r e t a t i o n o f " p r o b a b i l it 6 s d 6 n o m b r a b l e s " . I n p a r t i c u l a r, i t d ec i si v e ly s u st a in s

o u r a s s e r t io n t h a t " c o m p o s i t e p r o b a b i l i t y " l a y a t t h e h e a r t o f h i s t h e o r y , a n d t h a t

c o u n t a b l e a d d i t i v i t y ( a n d i t s c o r o l l a ry , c o u n t a b l e s u b - a d d i t i v i t y fo r n o n - d i s j o i n t

u n i o n s ) w a s i n n o w a y c e n t r a l t o B O R EL 's c o n c e p t i o n o f p r o b a b i l i t y . B O R E L (1 91 2)

a l so i n d i c a te s h o w f a r BO R EL w a s f r o m a p p l y i n g " C A N T E L L I " r e a s o n i n g t o t h e c a s e

~ p , c o n v e r g e n t : i n d e e d , t o s h o w h i s (1 91 2) m o d i f i c a t i o n o f h is Z e r o - O n e L a w

v a l i d a t e s t h e C o n t i n u e d F r a c t i o n T h e o r e m ( b o t h c a s e s ) , h e c h o s e t h e c o n v e r g e n t

c a se f o r d e t a i l e d e x p o s i ti o n , a l t h o u g h t h e " C A N T E L L I " r e a s o n i n g (i.e. u s e o f s ub -

a d d i t i v i t y ) p r o v i d e s a s t r o n g e r a n d s i m p l e r m o d i f i c a t i o n fo r th i s c as e .

F i n a l l y , h e m a r r e d h i s o w n d e f e n s e b y i n s i s t in g t h a t a s p e c if ic in e q u a l i t y o f

BOREL (1909) w as e qu iva le n t to one o f BERNSTEIN 'S (1911) . In f ac t wh i l e the de s i re d

i n e q u a l i t y w a s p e r h a p s l a t e n t i n t h e r e a s o n i n g o f BO R E L 's p a p e r o f 1 9 0 9 , o n l y a

h o p e l e s s l y w e a k e n e d v e r s i o n w a s e x p l i c i t l y g i v e n i n t h e p a s s a g e c i t e d b y B O R E L .

Axiom atic Prob ability 165

T h e p a p e r o f 1 91 2 p a r t i a l ly r e b u t s t h e p o s s i b l e c l a i m t h a t B O R EL c o u l d p e r f e c t l y

w e l l h a v e p a t c h e d u p h i s p a p e r o f 1 90 9 if t h e n e e d h a d b e e n p o i n t e d o u t t o h i m .

I n d e e d , h e h a d t h e o p p o r t u n i t y i n 1 9 12 , b u t t h e r e f a il e d to a l t e r h is o r i g i n a !

c a l c u l a t i o n s s u f f i c i e n t l y .

8 .2 . T h e C r i t ic i s m : T h e C o n t r i b u t i o n o f F . B e r n s t e i n ( 1 9 1 1 )

W r i t i n g i n 1 91 1, F . B E RN S TE IN c o n s i d e r e d a p r o b l e m i n C e l e s ti a l M e c h a n i c s

w h i c h r e p r e s e n t e d o n e o f t h e s e v e r a l e ar l y in s t an c e s o f t h e i n t r u s i o n o f " p o i n t s e t

t h e o r y " i n to d y n a m i c s a n d c la s si ca l m e c h a n i c s . T h e p r o b l e m c o n s i d e r e d i s: w h i c h

p o s s i b le c o n f ig u r a t i o n s o f a ce r t a in 3 - b o d y p r o b l e m a d m i t t e d a " m e a n m o t i o n " .

P u t e n t i r e l y i n m a t h e m a t i c a l t e r m s , a n d d e a l i n g f i r s t w i t h t h e n - b o d y p r o b l e m ,

B E RN ST EIN c o n s i d e r e d t h e r e a l a n d i m a g i n a r y p a r t s o f t h e f i ni te s u m

2 rme (g'~t+hm) ( rm>0) .1

T h e e x i st e n ce o f a m e a n m o t i o n m e a n s , f o r BERNSTEIN, t h a t t h e s u m m a y b e p u t in

t h e f o r m

r( t) e i~(O

w h e r e

c o (t ) = c t + f ( t )

a n d f ( t ) is b o u n d e d f o r a l l t. T h e c o n s t a n t c i s t h e n c a l l e d t h e m e a n m o t i o n , a n d c

= l i m c o (t ). ( T h e r e q u i r e m e n t t h a t f ( t ) b e b o u n d e d w h i c h w a s o f in t e re s t t o

BERNSTEIN, h a s s i n ce b e e n d r o p p e d f r o m t h e d e f in i ti o n o f m e a n m o t i o n , b u t w e u s e

t h e t e r m a s e m p l o y e d b y BERNSTEIN.)

I f n = 2 , a m e a n m o t i o n e x i st s , a n d i f n = 3 a n d o n e o f t h e r l , r2 , r3 e x c e e d s t h e s u m

o f t h e o t h e r t w o , a g a i n a m e a n m o t i o n e x is ts . T h i s w a s s h o w n b y L A G R A N G E . I f n

= 3 a n d L A G R A N G E ' s c o n d i t i o n f a il s, t h e q u e s t i o n r e d u c e s , f o l l o w i n g B O H L (1 90 9),

t o a d i s c u s s i o n o f t w o a s s o c i a t e d q u a n t i t i e s , p , ~, d e f i n e d b y

g 2 - - g l 1P = - - ~ = - ( ~ 3 + Pc o2 ).

g 3 - g l ' rc

H e r e t h e n o t a t i o n is d e f i n e d a f t e r o b s e r v i n g t h a t , s i n c e L A G R A N G E ' s c o n d i t i o n f ai ls ,

t he r e i s a ( un ique ) t r i an g l e w i th s i de s r l , r 2 , r 3 ; coa , ( o2 , co3 a r e de f ined a s t h e ang l e s

o p p o s i t e r l , r2 , r3 r e s p e c t i v e l y . T h e r e s u l t o f B O H L in 1 90 9 w a s t h a t l i m c o (t) e x i s ts

f o r e v e r y c h o i c e o f p , ~, a n d s o ' ~ ~ t

c o(t) = c r + f ( t )

w h e r e l i m f ( t ) = 0 . B u t o n t h e o t h e r h a n d , h e s h o w e d t h e r e is a d e n s e s e t o f p o i n t st ~ o 0 t

( p, ~) i n t h e u n i t s q u a r e f o r w h i c h t h e c o r r e s p o n d i n g f ( t ) is n o t b o u n d e d , a n d h e n c e

f o r w h i c h t h e r e i s n o m e a n m o t i o n i n t h e s e n s e e m p l o y e d b y B E R N S T E I N .

16 6 J. BARONE 8,: A. NOVIKOFF

BERNSTEIN'S paper is devoted to inves t igat ing the m e a s u r e of the poi nts (p, {) in the

uni t sq uare for which a mea n mo t io n ex is ts . His r e su l t is tha t th i s se t i s o f me asu re

The s ign i f icance of th i s f o r Ce les ti a l M echa nics i s no t o f in te r es t he r e .

BERNSTEIN shows how t he r e s u lt r e duc e s t o de t e r m i n i ng t he m e a s u r e o f po i n t s i n

t he un i t in t e r va l wh i c h pe r m i t a p p r o x i m a t i o n t o h i gh de g re e by m e a n s o f r a t i ona l

app rox im ant s . Because of th is , BERNSTEIN i s l ed to d i scuss the m easu re of

( i rr a t iona l ) num be r s i n t he un i t i n t e r va l whos e c on t i nue d f r a c t ion e xpa ns i ons ha ve

inf in i te ly m an y e lem ents an which a r e l a rge . Spec i fica lly , he es tab l i shes and em plo ys

t he r e s u lt t ha t t he s e t o f x who s e c o r r e s pond i ng e l e m e n t s a , = a , ( x ) are un-

bou nde d a s n r a nge s ov e r the od d i n t e ge r s is o f m e a s u r e 1.

C onve n i e n t l y , a ll t he t he o r e m s a b ou t c on t i nue d f r a c ti ons t ha t he e s ta b l i she s a r e

g r ou pe d i n a s e l f -c on t a i ne d s e c ti on e n t i t le d " T he ge om e t r i c p r oba b i l i t y f o r t he

a p p r o x i m a t i o n o f r e al n u m b e r s b y r a t i o n a l n u m b e r s , t o s t ro n g e r o r d e r t h a n

c on t i nue d f r a c t ion a pp r o x i m a t i ons , a nd r e l a t e d t op i c s " . T he c h i e f r e s u lt e m p l o ye d

f o r a p p l i c a ti o n s o f m e a n m o t i o n s is hi s T h e o r e m 2 :

Those i rrat ional numbers x in (0, 1) which sat i s fy

a~ <k o r a n r ~ k , k > l

f o r r= 0 , 1, 2 , . . . along so me spec i f i ed subsequ ence n 1 < n 2 < n 3 < . . . < n ~ . .. are

poin t s o f measure zero . BERNSTEIN (1911: 428).

Ta kin g the un io n of these "N ul l -m en ge " for k - - 1, 2 , 3 . . . c lea r ly impl ies the r esu l t

ab ou t un bo un de d gro wt h of a , wi th od d ind ices c i t ed above . BERNSTEIN fur the r

e s t a b l is he s t he B OR EL C on t i nue d F r a c t i o n T h e o r e m i n a ne w manner (BERNSTEIN

( 1911: 256) T he o r e m 4). H i s p r o o f is une xc e p t i ona b l e , a l t houg h h is s um m i ng up o f

t he a r gu m e n t i n the f o r m o f a t he o r e m i s c l oudy . He i s a t pa i n s t o po i n t t o t he f l a wed

cha rac te r o f BOREL' s pr oo f in i ts use of inde pend ence be tw een " t r i a l s " a ,__>q~(n),

a nd ne ve r t o u s e s uc h r e a s on i ng h i m s e l f .

We n ow ske tch BERNSTEIN's sec t ion 2 . To the n o ta t io n a l r eady in t roduc ed , we

a d d t h e n o t a t i o n

P [ a ,> = k l a l = m i , a 2 = m 2 . . . . . a , _ l = m , _ l ]

t o de no t e t he condi t i ona l probab i l i ty t ha t a , ( x ) > k g i ve n t ha t ai (x ) = m~, i = 1, 2 .. .. ,n - 1. This i s, by de f in i t ion , a r a t io of l engths : the nu m era to r i s

l [ al = m i , . .. , a , - l = m , - 1 , a , = m , ] ,m n ~ k

t h e " l e n g t h " (i.e. m e a s u r e ) o f t he po i n t - s e t

[a I =m 1, . . . , an_ 1 = mn_ 1, a , = m n ]m n ~ k

a n d t h e d e n o m i n a t o r i s

l [ a i = m l . . . . . a , l = m , - i ] ,

the l ength of a sing le in te rva l co r r esp ond ing to the se t in which a i ( x ) = m i ,

i = 1 , 2 , . . . , n - 1.

Axi om a t i c P roba b i li t y 167

B E R N S T E IN c al l s t h i s r a t io t h e " g e o m e t r i c p r o b a b i l i t y " t h a t a , > k. H i s r e s u l ts

a r e a l l d e p e n d e n t o n t h e p r e l i m i n a r y i n e q u a l i t i e s :

2 /~ k l a x = m s , . .. , a . _ s = - m ~_ l ] < ~ ,

( 8 . 1 )

1 - k - - ~ < P [ a , < k l a s = r n s , . ,a n_ l = m , _ / l < l - l k . I

E q u a t i o n s ( 8 . 1 ) a r e B E P ,NSTEI N ' s ( 1 9 1 1 : 4 2 6 : ( 4 2 ) a n d ( 4 3 ) ) r e - wr i t t e n i n t h e

t e r m i n o l o g y o f c o n d i t i o n a l p r o b a b i l i t y . ( T h e c o r r e s p o n d i n g " g l o b a l " i n e q u a l i t ie s

~ - k ] <~ [ -+ l '

1 - k ~ - f . < P [ a " < k ] < l - ~ ' 1 (8.2)

f o l lo w i m m e d i a t e l y a s w e i g h t e d a v e r a g e s o f t h es e .)

T h e i n e q u a l i t i e s ( 8 . 2 ) o n g l o b a l p r o b a b i l i t i e s ( c f . BERNSTEI N ( 1 9 1 1 : 4 2 7 :

e q u a t i o n 4 7)) m a y b e c o m p a r e d t o t h e s o m e w h a t a n a l o g o u s i n e q u al it ie s o f B O R EL :

2 33 (k + 1~ k ] < k + 2 ' (8 .3 )

k - 1 3 k + 1

k + 2 < P [ a "< = k ] < 3 k + 3 (8.4)

( c f ( 8 . 1 ) a n d i t s c o m p l e m e n t e d f o r m ) .

T h e d i s t i n c t i o n b e t w e e n t h e s e a n d B E R N S T E IN 's i n e q u a l i t i e s (8 .1 ) i s c r u c i a l :

B O R E L h a d n o t c a l c u l a t e d b o u n d s f o r c o n d i t i o n a l p r o b a b i l i ti e s , b u t o n l y f o r

" g l o b a l " p r o b a b i l i ti e s . ( E v e n w h e r e B O R EL c a lc u l a t e s b o u n d s f o r r a t i o s o f g l o b a l

p r o b a b i l i t i e s , e . g .

P [ a , > k + l ] P [ a l = m s . . . . a ~ _ l = m , _ s , a , > k + l ]o r

P [ a , > k ] P [ a l = m s . .. . , a , _ l = m , _ s , a n > k] '

c f. (6 .1 ) a n d ( 6.2 ), t h e s e a r e n o t p r o b a b i l i t i e s o n a n c o n d i t i o n e d o n t h e b e h a v i o r o f

e a r l i e r a s , . . . , a , _ s , i . e . a r e n o t B E R N ST E IN 'S " g e o m e t r i c p r o b a b i l i t i e s " .)

B E R N S T E I N t h e n e m p l o y s a n i n g e n i o us a r g u m e n t t o s h o w t h a t th e p r o b a b i l i ty

o f c y l i n d e r s e ts d e s c r i b e d b y th e s i m u l t a n e o u s c o n d i t i o n s

an~ > k~ ,, a~2 > kn~ ... , a~r > kn~

s a t i s fy i n e q u a l i t i e s

1 1 1k nl kn2 kn r " k n l 'a n 2 > k . . . . . . a ~ r > k J

< ( k . l + 1 ) ( k . r + 1 )

( 8 . 5 )

168 J. BARONE • A. NOVIKOFF

a n d s i m i l a r l y

(1 k 2 + l ) . . . ( 1 - k - - ~ ) < P [ a , l < k . . . . . , a , r < k . r ]r

T h e s e b o u n d s a r e f o r " g l o b a l ' , n o t c o n d i t i o n a l p r o b a b il i t ie s . M o r e i m p o r t a n t , t h e y

a r e f o r t h e p r o b a b i l i t i e s o f a p a r t i c u l a r k i n d o f c y l i n d e r s et , n a m e l y i n t e r s e c t i o n s o f

t h e i n d i v i d u a l s e ts (o r " t ri a l s " ) d e f i n e d b y i n d i v i d u a l i n e q u a l i ti e s o f t h e f o r m a , > k ,

( o r a , < k , , r e s p e c t i v e l y ) , fo r v a r i o u s f i n it e c h o i c e s o f n . M o s t i m p o r t a n t o f a ll , t h e s e

i n e q u a l i t i e s c o i n c i d e w i t h w h a t o n e c o u l d h a v e o b t a i n e d f r o m t h e i n d i v i d u a l

" g l o b a l " i n e q u a l i t i e s

1 2~ = k , J < k , + l

2 11 - < P [ a , < k , ] < l - - -

k , + 1 k ,

( e s se n t ia l ly e q u i v a l e n t t o t h e " g l o b a l " i n e q u a l i ti e s o f B O R E L ; c f (8 .3) , an d (8 .4)) h a d

o n e b e e n a l l o w e d t o a s s u m e i n d e p e n d e n c e a s w e l l .

B E R N ST E IN 'S i n g e n i o u s a r g u m e n t f o l lo w s f r o m t h e p u r e l y a l g e b r a i c i d e n t i t y

P [ a i = m a , a2 ~m2 , . . . , an =toni

= P [ a , = m , l a i= m i , l _ < i < n - 1 ] P [ a , _ i = m , _ i j a i = m i , l _ < i < n - 2 ] (8.7)

• . P [ a 3 = m 3 1 a l = m i , a z = m 2 ] P [ a2 = m z l a i = m l ] P [ a l = m l J .

T h i s i s t h e f o r m o f " p r o b a b i l i t 6 s c o m p o s 6 e s " w h i c h is a d e q u a t e l y g e n e r a l f o r t h e

c a s e a t h a n d . T h i s m i g h t w e l l b e c a l l e d t h e ( f i n i t e ) C h a i n L a w o f P r o b a b i l i t y .

T o p r o v e B E R N S T E I N 'S i n e q u a l i t i e s ( 8 .5 ) a n d (8 .6 ) f r o m t h e C h a i n L a w , d e n o t e

t h e p r o d u c t o n t h e ri g h t o f t h e C h a i n L a w b y I ~ ( m i , m 2 . . . . . m , ). F i xtl

m l , m 2 , . . . , m , _ 1 a n d s u m b o t h s id e s fo r al l v a l u e s o f r e, r a n g i n g f r o m a f i x ed l o w e r

l i m i t t o i n f in i ty . I f t h e l o w e r l i m i t i s 1, b o t h s i d e s s im p l i f y , a n d t h e r e s u l t i s t h eo r i g in a l a s s e r t io n w i t h a l l r e f e re n c e s t o a , d e l e te d , i n v o l v in g I ~ o n t h e ri g h t - h a n d

s id e . I f t h e l o w e r l i m i t i s s o m e i n t e g e r k , > 1, t h e l e f t - h a n d s i d e b e c o m e s

P [ a l = m l , . .. , a , _ i = m , _ l , a , > k , ]

w h i l e th e r i g h t - h a n d s i d e o f (8 . 7 ) c a n b e e s t i m a t e d f r o m a b o v e a n d f r o m b e l o w a s t h e

p r o d u c t

I-[ = P [ a , _ i = m , _ i I a i = m i , 1 < - - i< _ n - 2 ] . .. P [ a 2 = m z l a i = m i ] P [ a i = m a ]n- - i

t i m e s t h e u p p e r a n d l o w e r e s t im a t e s , re s p e c t iv e l y , w h i c h h o l d f o r th e l e a d i n g f a c t o r

(el. (8.1))

P [ a , = m , [ a i = m i , 1 < - i< _ n - l J .

I n o t h e r w o r d s

1 2k , , -1 [ I k ,]<--= k , + l , - 1 1 ~ '

I n a s i m i l a r f a s h i o n w e e s t i m a t e [ I f r o m a b o v e a n d b e lo w , o r r e d u c e i t t o ~ I ,n - - 1 n - - 2

a s fo l l o w s : W e s u m o v e r m , _ 1 f r o m s o m e l o w e r l i m i t t o in f in i ty , d e l e t in g r e f e r e n c e

t o a , 1 i f t h i s l o w e r l i m i t is 1, o r e l s e o b t a i n i n g

1 2k° f I < I I < f I- - n - - 2 n - - 1 k n 1 + 1- n - - 2

i f t h e l o w e r l i m i t i s k , _ 1 > 1. P r o c e e d i n g i n t h is f a s h i o n , u s i n g a p r e c r i b e d f in i te

c o l l e c t i o n o f i n d i c e s n 1 . . . . n r w i t h p r e s c r i b e d l o w e r l i m i t s k ,~ , k ,2 . . . . , k , r , w e a r r i v ea t t h e d e s i r e d i n e q u a l i t y (8 .5 ). T h e m i s s in g i n t e r m e d i a t e i n d ic e s c o r r e s p o n d t o

s u m m a t i o n s w i t h l o w e r l i m i t 1. T h e i n e q u a l i t y (8 .6 ) f o r P [ a , ~ < k . . . . . . a , r < k j i s

o b t a i n e d s i m i l a r l y or , a l t e r n a t i v e l y , c a n b e v i e w e d a s a n i m m e d i a t e c o r o l l a ry .

B E R N S T E IN n e x t e x p l i c it ly e m p l o y s t h e c o n c ep t o f m e a s u r e a n d t h e va r i ou s f o r m s

o f c o u n t a b l e a d d i t i v i t y ( r e f e rr i n g t o L E B E S G U E ( 19 0 6) ) i n o r d e r t o e x t e n d p r e v i o u s

c o n s i d e r a t i o n s t o s e t s d e f i n e d b y i n fi n it e ly m a n y i n e q u a l it ie s . H e t h u s c a l c u l a te s

t h e p r o b a b i l i t y o f t h e s e t

[ a ,~ > k , 1 , a , 2 > k , ~ , . . . , a , > k , ~ , . . . ]

a n d o f t h e s e t

[ a,~ < k n~ , a , ~ < k , ~ , . . ., a , < k . . . . . . ] .

T h e s e s e ts i n v o l v e in f i n it e s e q u e n c e s o f i n e q u a l i t i e s a n d t h u s a r e n o l o n g e r c y l i n d e r

s e ts . F o r i n s t a n c e , h i s T h e o r e m (2 ), r e f e r r e d t o a b o v e , r e s u l ts b y a p p l y i n g

B E R N S T E I N ' s i n e q u a l i t i e s ( 8. 5) a n d ( 8 .6 ) t o a g i v e n s e q u e n c e { n, } o f i n d i c e s , a n d

c h o o s i n g k ,~ = k f o r r = 1 , 2 , 3 , . . . . S i n c e

l i m ( l _ k ~ ) = l l m ( 1 _ ~ ) ~ = l i m ( ~ )r = l i m 1

i n e q u a l i t i e s ( 8.5 ) a n d ( 8.6 ) a n d c o u n t a b l e a d d i t i v i t y i m p l y t h e t w o r e s u l t s :

P [ a , i > k , a , 2 > k . . . . . a , r > k , . .. ]

= l i m P [ a , 1 > k , a ,2 > =k , . . . , a , > k ] = 0r~oo

P [ a , 1 < k , a ,2 < k , . . ., a , . < k, . . . ] = l i m P [ a , 1 < k , . . . , a , . < k ] = 0 .r ~ c o

T h e s e t s [ a , . > k , r = 1 , 2 , 3 . . . ] a n d [ a , r < k , r = 1 , 2 , 3 , . . . ] a r e b o t h o f m e a s u r e z e r o

f o r e a c h v a l u e o f k . H e n c e u n i o n s o f s e ts s u c h a s k = 2 , 3 , . . . a r e s t il l o f m e a s u r e z e r o ;

t h u s t h e p r o b a b i l i t y t h a t { a ,r } is b o u n d e d f r o m a b o v e ( o r b e l o w ) i s z e r o .

T o o b t a i n B O R E L'S t h e o r e m o n C o n t i n u e d F r a c t i o n s , i t s u ff ic e s t o a p p l y ( 8.5 )

a n d ( 8 .6 ) t o t h e i n d i c e s n , n + 1 . . . , n + r , a n d c h a n g e k , , k , + 1 . . . , k , + r t o t h e B O R E L

170 J. BARONE & A. NovIKOFF

n o ta ti o n qS(n), qS(n+ 1 ), . . . , qS(n + r). Th en fro m ine qu ali t y (8.6) i t follows t ha t

1 ¢ (v ) -+ l < P [ a v < ( ~ ( v ) ' v = n ' n + l . . . . . r ] < I ] - •v = n v ~ n

If ~ 11 0 (~ d i ve rge s ,

P [ a ~ < O ( v ) , v = n , n + 1, . . . ] = l im Pl-a~ < qS(v), v = n , n + 1 . . . . r] = 0r ~ 3

1(the diver gent case of BOREL's theor em) . I f ~ ~ conve rges , (8.5) assures that

P[a~ < O(v) , v= n , n + 1, . . . , r]

i s pos i t ive (and less than 1) and in fact l ies between

( ) a n d f i 1 - ~fi 1 q~(v)+l-

~ n v = n

To achie ve BOREL's resul t in this case , i t i s nece ssary to co m pu te the p rob ab i l i ty

tha t a~ < ~b(v) hold " f r om some n on" , tha t i s the pro babi l i ty of the unio n of the se ts

[av< ~b(v ) ,v> n] over a l l in teger va lues of n . Thi s pro babi l i ty i s thus 1, aga in

va l ida t ing BOREL's resu l t. I f the phrase " f r om some n on" is in te rp re ted to me an" f r o m s o m e f i x e d v a l u e of n on " then the answer i s s t r ic t ly l es s than one . Thi s

amb igui ty o f l anguage resu l t ed in needles s , and for our purpose , i r re l evant conf l i c t

be twe en w ha t BOREL sa id he pr ov ed a nd wha t BERNSTEIN ac tua l ly succeede d in

proving. Their resul ts coincide, but BERNSTEIN actual ly suppl ied a val id,

i nde pe nde n t l y c onc e i ve d p roo f , e xp li c it ly u ti li z ing t he l a ngua ge a nd t he o re ms o f

mea sure theory . BOREL, by cont ras t , f ir s t a s se r t ed the th eor em in 1909 but the re

s upp l i e d a non -p roo f , v i t i a te d by de pe nde nc e be t w e e n " t r i a l s " a nd e mpl oy i ng on l y

t he Z e ro -O ne L a w . A s w e ha ve s e e n , t he Z e ro -O ne L a w p rove d by B O R E L ha s

noth ing to d o wi th the l angu age of LEBESGUE measure , and , in B O R E L ' S expos i t ion ,

has no c lea r ly recog nized l ink to co unta ble addi t iv i ty or sub-addi t iv i ty .Af te r achieving h i s pr oo f of BOREL'S resu l t on Con t inu ed Frac t ions , BERN-

STEIN emphas izes tha t the indiv idua l event s a n = m . a re no t i nde pe nde n t . H e

c ompu t e s t he fou r p roba b i l i t i e s P [ a 1 = 1], P [ a 1 = 2 ] , P [ a I = 1, a 2 = 1], P [ a 1 = 2 ,

a 2 = 1] as lengths of intervals , and obser ves tha t the tw o ra t ios

P [ a 1 = 1, a 2 = 1 ] a n d P [ a l = 1]

P [ a l = 2 , a 2 = 1 ] P [ a 1 = 2 ]

a r e une qua l , a s t r ue i nde pe nde nc e w ou l d r e qu i r e . H e po i n t s ou t t ha t t he a de qua t e

law of "probabi l i t6s com pos6 es" (or a s BERNSTEIN ca ll s it "da s Th eo rem der

z us a mm e nge s e t z t e n W a hr s c he i n l i c hke i t e n" ) invo l ves p rodu c t s o f compos i t e p r o b a -

b i li ti e s to co mp ute p robabi l i t i e s o f the s imul tan eous occu r rence of severa l events .

Thi s i s wha t we have ca l l ed the Cha in Law. The example he g ives , in modern

nota t io n , cons ide rs two event s , A and B , where A - - • Ai deco mp oses event A in to

i ts poss ible a l ternat ives . Then

P(A c~ B) = P (@ (A i c~ B)) = ~ P(A i) P( B ]Ai).

I f q < P (B I Ai) < ~1for all i , then

q P(A ) ~ P(A ~ B) __<qP(A) .

Thus , bo und s on c ondi t io na l prob abi l i t i e s for B , va l id for a ll a l t e rna t ive condi t ions ,

g ive r i se to the same inequ a l i ty concern ing P(A ~ B) as wou ld hav e been ob ta ined i f

the g loba l probabi l i ty P (B) were known only to sa t i s fy

q< P(B)<~I

and the event s A and B were independent .

Thi s obse rva t ion could have been appl i ed d i rec t ly to BOREL's Zero-One Law

i n t he de g re e o f ge ne ral i ty o f " de nu me r a b l e p roba b i l i t y " . S inc e, how e ve r ,

BERNSTEIN'S w hol e v i e w po i n t is no t t o c r e a t e a ne w t he o ry o f de num e ra b l e

p roba b i l i t y bu t t o e m pl oy t he e x i st ing t he o ry o f me a s u re , he doe s no t m a ke t he

observa t ion tha t we can make on h i s beha l f : BERNSTEIN in t roduced the key

e lement in genera l i z ing the BOREL Zero-One Law in the case not cons ide red by

CANTELLI, i.e. the d ive rgent case . CANTELLI dem oted the h ypothes i s of inde-

pendence in BOREL's Zero-One Law in the convergent case ( the "ze ro" ha l f ) by

showing i t could be omit ted. BERNSTEIN general ized the case of divergence ( the" o ne " ha l l) by s how i ng t he hypo t he s i s o f i nde pe nde nc e c ou l d be w e a ke ne d t o

requi r ing adequa te (upper ) bounds on ce r t a in condi t iona l probabi l i t i e s , these

bounds themselves forming a divergent ser ies . In fact , had he been suff ic ient ly

thoro ugh -go ing in h is use of measu re the ory , and had he no t ins i st ed , like BOREL,

on t rea t ing the conv ergen t and d ive rgent cases in like man ner , he cou ld have show n

tha t in the conv ergen t case sub-addi t iv i ty es t imates a lone w ould h ave es tab l i shed

the theo rem in te rms of the g loba l probabi l i t i e s P[an > ~b(n)]. He wou ld thus hav e

ant ic ipated CANTELLI. Indeed, in his sect ion (1) , he expl ic i t ly calcula tes the

mea sure of the l imi t in fe r ior of a co l l ec t ion of set s, a t t r ibu t ing th is ca lcu la t ion to

LEBESGUE (1906). This is , in fact , a m od er n "CANTE LLI-l ike" calcula t ion . Ho we ve rthat may be, the fact i s that BERNSTEIN gave the first valid proof of BOREL'S

result on Continued Fractions, and gave the first generalization of BOREL'S Zero-

One La w to cover the possibility of dependence.

8.3. Borel's Response: Borel (1912)

After BERNSTEIN'spaper , the s t a tus of BOREL'S Con t inue d Fra c t io n T he ore m

was, for a short per iod, in doubt . Both BERNSTEIN and BOREL agreed that the

prob abi l i ty tha t a , < qS(n) should hold f rom some n on was ze ro i f ~ d ive rged .

BOREL asser ted this in the form that the p rob abi l i ty t hat a , > qS(n) hold s inf ini te ly

1often, the Aoo of his pa pe r of 1909, is 1. In the case ~ ~ con ve rge nt, BOREL

172 J. BARONE8z; A . NOVIKOFF

a s s e r t e d t h a t t h e p r o b a b i l i t y t h a t a n > ~ b ( n ) h o l d s i n f i n i t e l y o f t e n i s z e r o ; t h u s

a n < ~b(n) h o l d s f r o m s o m e n o n w i t h p r o b a b i l i t y o n e . B E R N S T E I N a s s e r t e d i n t h e

c o n v e r g e n t c a se t h e p r o b a b i l i t y t h a t a n < g ) ( n ) h o l d s f r o m s o m e n o n ( m e a n i n g

a n < ~ b(n ), a n + 1 < ~ b( n + 1 ), . . . f o r a g iv e n n ) i s p o s i t i v e b u t l e s s t h a n o n e . BE R N S T E I Nt h u s c r i ti c i z ed b o t h B O R E L'S a r g u m e n t a n d h i s r e su l t. S t u n g b y t h is , B O R E L

r e s p o n d e d w i t h a n o t e (B O RN E ( 19 1 2) ). T h e r e h e i n s i s te d t h a t h i s r e s u l t w a s c o r r e c t

a s s t at e d , n a m e l y A ~ is z e r o o r o n e i n t h e c o n t i n u e d f r a c t i o n c a s e a c c o r d i n g a s

1~ c o n v e r g e s o r d i v e rg e s , b u t c o m p l e t e l y r e w o r k e d th e p r o o f i n s u c h a f a s h i o n

t h a t d e p e n d e n c i e s w e r e pe r m i t te d . H a v i n g g e n e r al i z ed h i s o r i g in a l Z e r o - O n e L a w

i n t h is w a y , B O R E L a t t e m p t e d t o a p p l y i t t o p r o v e h i s c o n t i n u e d f r a c t i o n r es u lt . A s

w e s h a ll s ee , t h e a p p l i c a t i o n r e q u i r e d a m o d i f i c a t i o n o f h i s o ri g i n a l c a l c u l a t i o n s o f

1 90 9. T h i s B O R E L f a il e d to u n d e r t a k e , p e r h a p s u n w i l l i n g t o c o n c e d e s u c h a n

i n a d e q u a c y i n h i s c a l c u l a t io n s o f 1 90 9.

B O R E L ' s n e w p o o f o f t h e g e n e r a l i z e d Z e r o - O n e L a w e s s e n t i a ll y c o i n c i d e s w i t h

t h e a r g u m e n t o f B E RN S TE IN . T h a t is, th e g e n e r a l l a w o f " p r o b a b i l i t ~ s c o m p o s 6 e s "

( in t e r m s o f c o n d i t i o n a l p r o b a b i li t ie s ) , r e p l a c e s c o u n t a b l e i n d e p e n d e n c e a s t h e

e s s e n ti a l to o l . B O R E L t h e n s h o w s t h a t a p p r o p i a t e i n e q u a l it i e s o n c o n d i t i o n a l

p r o b a b i l i t i e s g i v e r is e t o " i n d e p e n d e n c e - l i k e " e s t i m a t e s i n th e f o r m o f p r o d u c t s (cf.

(8 .5 ), (8 .6 ), a b o v e ) . B u t B O R E L , i n 1 9 1 2 , p l a c e s t h e s e a r g u m e n t s i n t h e g e n e r a l i t y o f

h i s o r i g i n a l Z e r o - O n e L a w , i.e., t h e s p a c e o f a ll d e n u m e r a b l e s e q u e n c e s o f t ri a ls ,

w i t h p o s s i b l e s u c c e s s o r f a i l u r e a t e a c h t ri a l, w h e r e a s B E R N S T E I N h a d b e e n

c o n c e r n e d o n l y w i t h t h e a p p l i c a t i o n t o c o n t i n u e d f r a ct i o ns .

B O R E L ' s o r i g i n a l n o t a t i o n Pn f o r s u c c e s s a n d qn f o r f a i l u r e a t t h e n th t r i a l i s

i n a d e q u a t e f o r h is n e w p r o o f. W e a l s o n e e d t h e c o r r e s p o n d i n g c o n d i t i o n a l

p r o b a b i l i ti e s , s i nc e w e e x p l i ci t ly a l lo w d e p e n d e n c e . L e t u s i n t r o d u c e t h e n o t a t i o n

s , f o r a " s u c c e s s " p a r a m e t e r : s , = 1 m e a n s t h e n th t r i a l w a s a s u c c e s s, s n = 0 m e a n s

i t w a s a f a i lu r e . L e t x b e a n i n f in i t e s e q u e n c e o f t r i a l s ; t h e n s n ( x ) i s 1 i f x ha s

s u c c e s s a t t h e n th t r i a l a n d 0 i f x h a s f a i l u r e a t t h e n th t r i a l. T h e p r o b a b i l i t y t h a t a

s e q u e n c e h a s p r e s c r i b e d i n i t i a l v a l u e s s i = m l , i = 1 , 2 . . . . . n w h e r e e a c h m i = 0 o r 1 i s

g i v e n r e c u r s i v e l y b y

P E s 1 = m 1 . . . . s n = m , ] = P [ s , = m n Is 1 - - - - m I . . . . . S n - 1 = r a n _ i ]

. P [ S l = m l , . . . , s n _ l = m , _ l ]

s o t h a t"_L

P [ s 1 = m 1 . . . . , s , = r n , ] = I I P [ s k = r n k I s1 = / T / l , " ' , S k - 1 : l T l k - - 1 ] "

T h i s i s p r e c is e l y th e C h a i n L a w . S i m i l a r l y t h e p r o b a b i l i t y t h a t a s e q u e n c e f in i sh

w i t h t h e r e s u l t s S n + l = m n + p S n + 2 = m n + 2 . . . . g i v e n t h e i n i t i a l v a l u e s s l = m l , . . . ,

s , = m , is t h e i n f i n i t e p r o d u c t

f i P [ S , + k = m , + k l S l = r n l . . . . . s , + k - 1 = m , + k - 1 ]. (8 .8)k = l

T h i s i s t h e c o u n t a b l e e x t e n s i o n o f t h e C h a i n L a w t h a t B O R E L a s s er t s i n 1 91 2.

( B O R E L o f f er s n o j u s t i f i c a t i o n f o r t h e e x t e n s i o n f r o m t h e f in i t e t o t h e c o u n t a b l e c a s e ,

o n e m o r e i n s t a n c e o f a t a c it a n d e x t r e m e l y w e ll d i sg u i s ed d e p e n d e n c e o n c o u n t a b l e

a d d i t i v i t y . ) I t i s e m p l o y e d i n h i s r e s p o n s e t o B E R N S T E I N t o p r o v e t h e f o l l o w i n g

t h e o r e m : L e t A o , A s . . . . . A k . . . . b e , a s in 1 90 9, t h e p r o b a b i l i t y t h a t t h e r e a r e e x a c t l y

k s u c c e ss e s , a n d l e t A o~ = 1 - ( A o + A s + . . . + A k + . . . ) . L e t { p',} a n d {p', '} b e t w os e q u e n c e s s u c h t h a t

p ' , < P [ s n = l l s s = m l . . . . . S , _ l = m ~ _ s] < p~ '

f o r m l , m 2 , . . ., m , _ s r u n n i n g t h r o u g h a l l c h o i c e s o f 0 ' s a n d l 's .

I f ~ p " c o n v e rg e s , t h e n A , = 0 .

I f ~ p ' , d i v e rg e s , t h e n A , = 1.

E q u i v a l e n t l y ,

I f V p " c o n v e r g e s, t h e n A o + A 1 + . . . + A k + . . . . 1., n

I f ~P'n d i v e r g e s , t h e n A o + A I + . . . + A k + . . . . O.

B O RE L p r e s e n t s a p r o o f o n l y f o r t h e c o n v e r g e n t c a s e, s in c e t h a t is th e c a s e f o r

w h i c h B E R N S T EI N th i n k s ( e r r o n e o u s l y ) h i s r e s u l t i s i n c o n t r a d i c t i o n w i t h B O R E L 'S .

O f c o u r se "CANTELLI" r e a s o n i n g r e n d e r s BOREL's n e w p r o o f o f t h i s c a s e o b s o l e t e ,

b u t B O R E L 's p r o o f i n d i c a te s h o w t o h a n d l e t h e d i v e r g e n t c a s e a s w el l, a n d t h e r e f o r e

m e r i t s a t t e n t i o n .

I f ~ p ~ ' c o n v e r g e s , c o n s i d e r A o + A s + " " + A k . B O R EL s h o w s t h a t g i v e n e > 0 , k

c a n b e c h o s e n s o l a r g e t h a t

A o + . . . + A k > 1 - ~ .

T h e p r o o f is a s tr a i g h t f o r w a r d a p p l i c a t i o n o f t h e g e n e r a li z e d l a w o f c o m p o s i t e

p r o b a b i l i t i e s (8 .7 ): A 0 + . .. + A k is t h e p r o b a b i l i t y o f h a v i n g a t m o s t k s u c ce s s es ,

t h e r e f o r e g r e a t e r t h a n t h e p r o b a b i l i t y o f h a v i n g n o s u cc e ss e s f ro m t h e ( k + 1) st on . In

s y m b o l i c f o r m

A 0 + ' " + A k > P [ S k + 1 = 0 , Sk+ 2 = 0 . . . . ] .

B u t t h e l a t t e r is t h e w e i g h t e d a v e r a g e o f t h e p r o b a b i l i t i e s

P[Sk+ 1 = 0 , Sk+2 = 0 , . .. I S1 = m s . . . . , S k = m k ]

( cf. (8 .8 )) f o r a l l 2 k c h o i c e s o f ( m l . . . . , m k ). B y u s e o f t h e c o u n t a b l e e x t e n s i o n o f t h e

C h a i n L a w e a c h o f t h e s e p r o b a b i l i t i e s ( a n d h e n c e t h e i r w e i g h t e d a v e r a g e P [ sk + 1

- - 0 , S k ÷ 2 = 0 . . . . ] ) is e s t i m a t e d f r o m b e l o w b y t h e i n fi ni te p r o d u c t

( 1 - - p 'k '+ l ) ( 1 - - p 'k '+ 2 ) . . . I ~ ( 1 - - p ) ' ) .j = k + l

I f ~ p y < o% t h e n k c a n b e c h o s e n l a r g e e n o u g h t o m a k e t h is p r o d u c t g r e a t e r t h a n

1 - e, a s d e s ir e d , s h o w i n g

A o + A I + . . . + A k + . . . . 1.

174 J. BARONE8¢ A. NOVIKOFF

S i m i l a r l y , i f ~ p j d i v e r g e s , A o + A I + . . . + A k c a n b e s h o w n t o v a n i s h , t h o u g h

B O R EL o m i t s t h is . A p r o o f c a n b e b a s e d o n t h e i n e q u a l i t y

P I s ,+ 1 = 0 , s , + 2= 0 , . . . , S N = 0 I S 1 = m l , . . . , S n = t o n i < ( 1 - p ~ ,+ 1 ). .. (1 - p } ) ,

s u p p l e m e n t e d b y c o n t i n u i t y (i.e. c o u n t a b l e a d d i t i v i t y ) ; cf. BARONE (1974) .

I n s u m m a r y , B O R EL i n 1 9 12 p r e s e n t s a n e w , m o r e g e n e r a l Z e r o - O n e L a w , b a s e d

o n a n e w , m o r e g e n e r a l m e t h o d o f p r o o f ( b a s e d o n t h e c o u n t a b l e c h a in L a w u t il iz e d

p r e v i o u s l y a n d e x p l i c it ly b y B E RN S TE IN ). C o u n t a b l e a d d i t i v i ty a n d s u b - a d d i t i v i ty

a r e s ti ll n e g l e c te d p r i n c i p l e s ; c o u n t a b l e c o m p o s i t e p r o b a b i l i t y is e m p l o y e d i n th e

f o r m s

r~ o q r n l ~o

. . . . .L ' I ' J L ' I ' J ~ = 5

W e r e p e a t , th e e x t e n s i o n o f t h e C h a i n L a w f r o m t h e f i n it e t o t h e c o u n t a b l y i n f i n it e

c a s e i s o f f e r ed w i t h o u t e x p l a n a t i o n . B y c o n t r a s t , B E R N ST EIN u se s c o u n t a b l e

a d d i t i v i ty e x p l ic i tl y t o c a l c u l a t e t h e p r o b a b i l it i e s o f n o n - c y l i n d e r s et s o f th e f o r moo

( ~ E k , u t i li z i n g t h e a d d i t i o n a l i n s i g h t t h a t t h e s e p r o b a b i l i t i e s a r e ( in e v e r y c a s e1

u n d e r c o n s i d e r a t i o n ) m e a s u r e s o f m e a s u r a b l e s e ts .BOREL f ai ls to r e m a r k t h a t BERNSTEIN'sp r o o f is m u c h t h e s a m e a s t h e o n e h e

n o w p r e s en t s t o p r o v e t h e g e n e r a l iz e d Z e r o - O n e L a w , b u t a s se rt s t h a t " t h e n e w

p r o o f i s e s s e n t i a l l y t h e o n e t h a t w o u l d h a v e b e e n g i v e n i n 1 90 9 if a l l t h e c a l c u l a t i o n s

h a d b e e n w r i t t e n i n f u l l . "

I t o n l y r e m a i n e d f o r B O R E L t o r e a s s u r e t h e r e a d e r ( a n d h i m s e l f ) t h a t t h e

m o d i f i e d Z e r o - O n e L a w a p p li e s to t h e c o n t i n u e d f r a c ti o n ca se .

T h e n e e d e d i n e qu a l it ie s a r e t h e u p p e r a n d l o w e r b o u n d s f o r

P [ a ,> = ¢ ( n) l a l , . . . , a , _ l ]

i n d e p e n d e n t o f w h a t c o n s t r a i n t s a r e p l a c e d o n a l , . . ., a , _ 1 . B E RN S TE IN h a d f o u n d

e x a c t l y s u c h b o u n d s . B O R E L n o w i n 1 9 1 2 a s s e r ts t h a t t h e i n e q u a l i t ie s ( w h i c h h e

r e f er s t o a s " ( 2 3 ) a n d t h e r e a f t e r o n p a g e 2 6 8 " ) o f h i s p a p e r o f 1 90 9 p r o v i d e t h e s a m e

i n f o r m a t i o n . I n f a c t , t h e e s s e n t i a l i n e q u a l i t i e s o f B O R E L i n 1 9 09 a r e , i n t h e o r d e r o f

t h e i r d e r i v a t i o n ,

k P [ a . = k + l ] k + l- - <k + 2 P [ a . = k ] < ~ 3 '

2 P [ a . = k ] 6

k ( k + 1 ) < P [ a . = 1 ] < ( k + 1)(k + 2 ) '

~ P [ a . = k ] = 1,k = l

Axiom atic Probab ility | 75

3 k ( k + 1) < P l a n = k ] < ( k + 1 )(k + 2 ) '

2 3 k + l ]3 ( k + 1) = < k + 2 "

N o n e o f t h e s e in v o l v e c o n d i t i o n a l p r o b a b i l it i e s o f t h e s o r t r e q u ir e d . F o r e x a m p l e ,

t he l a s t i s no t

2 33 ( k + 1 ~ = k + l [ a x = m l . . . . , a n _ l = m , _ l ] < k + 2

a s w o u l d b e r e q u i r e d f o r a p p l i c a t i o n o f t h e g e n e r a l iz e d Z e r o - O n e L a w b u t i s o n l y a

m u c h w e a k e n e d v e r si o n o f it.

B O R E L r e m a i n e d a w a r e t h a t h i s o r i g i n a l e x p o s i t i o n w a s f la w e d , f o r i n 1 9 2 6 h e

p u b l i s h e d a c o n s i d e r a b l y m o r e d e t a i l e d a n d e x p a n d e d v e r s i o n o f h is p a p e r o f 19 09

u n d e r t h e t it le " A p p l i c a t i o n s ~t L ' A r i t h m 0 t i q u e e t/ ~ la T h 6 o r i e d e s F u n c t i o n s " , a s

f a s c ic u l e I o f T o m e I I o f h i s e x t e n s iv e " T r a i t 0 d u C a l c u l d e s P r o b a b i l i t6 s e t d e s es

A p p l i c a t i o n s " . N o w h e p r e s e n t e d t h e m a t e r i a l o f h is r e s p o n s e i n 1 9 1 2 t o

BERNSTEIN (i.e., h i s m o r e g e n e r a l Z e r o - O n e L a w ) b e f o r e t u r n i n g t o C o n t i n u e d

F r a c t i o n s . T h i s t i m e h e p r e s e n t s t h e a b o v e f iv e i n e q u a l i ti e s , n u m b e r e d (1 ), (2 ), (3 ),

(4), (5) , as s t a ted (i .e . , not c o n d i t io n e d ) a n d t h e n r e m a r k s :

T he i neq ua l i t i e s ( 1) , ( 2) , ( 3) , ( 4) , ( 5) r em a i n t r ue i f on e m ak es va r i ou s hy po t he s e s

c o n c e r n i n g t h e e l e m e n t s a l , a 2 . . . . a n 1. T h e s u m o f t h e l e n g th s lk, lk+ 1 o f t h ei n t e rv a l s c o n s i d e r e d , i n s t e a d o f ra n g i n g o v e r a l l p o s s ib l e v a l u e s o f th e e l e m e n t s

a l , . . . , a n _ 1 w i ll o n l y r a n g e o v e r t h o s e v a l u e s s a t i sf y i n g c e r t a i n g i v e n c o n d i t i o n s .

Th e g l ob a l p r o bab i l i t i e s P J an = k ] , P [ a n = k + 1 ] , P [ a , _> k + 1 ] w i l l be r e p l a ced

b y t h e p r o b a b il it ie s o b t a i n e d b y t a k in g a c c o u n t o f t h e h y p o t h e s e s m a d e o n t h e

i n it ia l n - 1 e l em e n t s , a n d w h a t e v e r t h o s e m a y b e , b y f o ll o w i n g t h e s a m e

r e a s o n i n g w h e r e b y t h e y w e r e e s ta b l i sh e d a b o v e t h e p r e c e d i n g i n e q u a l i ti e s w i ll

c o n t i n u e t o b e s a t i s f i e d b y t h e n e w p r o b a b i l i t i e s . B O R E L ( 1 9 2 6 : 6 6 ) .

Th i s s t a t em en t , i n 1926 , i s co r r ec t . BOREL 's a s s e r t i on , i n 1912 , t ha t h i s pape r o f

1909 a l r ea dy co n t a i n ed t he de s i r e d i ne qua l i t i e s i nv o l v i n g P'n an d p~' i s f a l s e o r , a t

l e a s t , d i s i n g e n u o u s .

8 .4 . E a r l y o b s e r v a t io n s o f L e b e s g u e a n d L ~ v y

O n e f in a l h i s to r i c a l c o m m e n t is in o r d e r c o n c e r n i n g B O R E L'S c o n t r i b u t i o n t o

t h e e x c h a n g e w i t h B E R N S TE IN . T h e s e c o n d e d i t i o n o f B O R E L 'S L e f o n s s u r l a T h d o r i e

d e s F u n c t i o n s (B O R E L ( 19 1 4 )) c o n t a i n s m a n y " n o t e s " a d d e d e s p e c i a l l y f o r t h i s

e d i t i o n . N o t e V c o n s i s t s o f a r e p r o d u c t i o n o f B O R E L ( 19 0 9 ) a n d B O R E L (1 91 2), in

to to . A f o o t n o t e w a s a d d e d t o t h is r e p r i n t e d p a p e r o f 1 9 12 w h i c h d o e s n o t a p p e a r i n

t h e o r i g i n a l . I t f o ll o w s t h e s e n t e n c e

W h a t i s v a l i d i n B e r n s t e i n ' s o b j e c t i o n i s t h a t t h e r e a s o n i n g t h a t I h a v e

g i v e n . . . a s s u m e s t h e p r o b a b i l i t i e s a r e i n d e p e n d e n t a n d s h o u l d b e m o d i f i e d

when t hey a r e no t . BOREL ( 1914 : 208 ) .

T h e f o o t n o t e s t a te s

I s h o u l d s a y t h a t t h i s o b j e c t i o n w a s m a d e t o m e i n a p e r s o n a l l e tt e r, b y

L e b e s g u e , a t t h e t i m e o f t h e p u b l i c a t i o n o f t h e Rendiconti. I a s s u r e d m y s e l f t h a t

t h e r es u l ts w e r e v a li d a n d a t t a c h e d n o i m p o r t a n c e t o t h e o b j e c t i o n ; I h a d e v e n

f o r g o t t e n i t w h e n , a f e w y e a r s l a t e r, I r e p l i e d t o B e r n s t e i n : o n l y a f t e r th e

p u b l i c a t i o n o f t h is r e s p o n s e , r e p r o d u c e d h e r e , d i d I c o m e a c r o s s th e o l d l e t t e r o f

L e b e s g u e . B O R E L ( 1 9 1 4 : 2 0 8 : F o o t n o t e ( 4 ) ) .

LEBESGUE and BERNSTEINw e r e n o t a l o n e i n o b s e r v i n g t h a t B O RE L 'S p a p e r o f

1 9 09 s u f fe r e d f r o m d e f e ct s o f e x p o s it i o n . I n a l e t t e r t o u s P . L g v Y w r o t e c o n c e r n i n g

Y e t , o n r e a d i n g i t, p e r h a p s w i t h o u t a t o n c e f u ll y u n d e r s t a n d i n g i ts

i m p o r t a n c e , m y i m p r e s s i o n w a s a b o v e a l l o n e o f s u r p ri s e.

I h a d n o i d e a t h a t s u c h s i m p l e p r i n c i p l e s , w h i c h h a d b e e n f a m i l i a r t o m e

s i n c e 1 90 7, w e r e n e w ( I a m s p e a k i n g o f t h e f ir s t t w o c h a p t e r s ; c h a p t e r 3 w h e r e

c o n t i n u e d f r a c t i o n s w e r e d i s c u s s e d w a s n e w t o m e ) .

I w a s s u r p r i s e d t h a t a s c h o l a r w h o s e w o r k o n d i v e r g e n t s e r i e s , e n t i r e

f u n c t io n s , P i c a r d ' s t h e o r e m , a n d t h e t h e o r y o f m e a s u r e I a d m i r e d , h a d g i v e n

s u c h c o m p l i c a t e d p r o o f s o f s u c h s i m p l e t h e o r e m s ( th i s t i m e I a m s p e a k i n g o f t h e

t h r e e c h a p t e r s ) . ( L e t t e r f r o m P . L E V Y , d a t e d D e c e m b e r 2 2, 1 9 69 .)

9. Early R e-workings o f Borel's Stro ng Law9.1. Introduction

T h i s c h a p t e r i s d e v o t e d t o o t h e r c o n t r ib u t i o n s f r o m t h e p e r i o d i m m e d i a t e ly

f o l lo w i n g B O R E L 's l a n d m a r k p a p e r o f 19 09 . T h e m a t h e m a t i c i a n s o f t h is t im e , w h o

w e r e a t t r a c t e d b y B O R E L ' s r e s u l t s , h e l p e d t o i l l u m i n a t e t h e r e l a t i o n b e t w e e n

m e a s u r e t h e o r y a n d p r o b a b i l i t y b y t h e i r e ff o rt s, t h o u g h t h e i r i n t e r e s t w a s p r i m a r i l y

i n t h e d i r e c t i o n o f t h e f o r m e r a n d n o t i n th e d i r e c t i o n o f p r o b a b i l i t y . T h r e e m e n

- -F A B E R , H A U S D O R FF a n d R A D E M A C H ER - - r e p r o v e d t h e B O R E L S t r o n g L a w

w i t h o u t a n y r e f e r e n c e t o t h e C e n t r a l L i m i t T h e o r e m ; i n d ee d , t h e ir m e r it ,

p a r a d o x i c a l l y , w a s t h a t t h e y d i d no t c o n c e r n t h e m s e l v e s w i t h t h e p r o b a b i t i s t i ci n t e r p r e t a t i o n o f t h e t h e o r e m . B y e x a m i n i n g t h e i r o r i g i n a l w o r k s , o n e c a n o b s e r v e

t h e s h i ft in v i e w p o i n t f r o m t h a t o f BO R EL (1 9 09 ) t o a v i e w p o i n t i n w h i c h p r o b a b i l i t y

( s p e c if i c a ll y g e o m e t r i c p r o b a b i l i t y o n [ 0 , 1 ] ) meant m e a s u r e . 1

F u r t h e r , H A U S D O RF F a l s o p r o v e d a r e s u l t o n c o n t i n u e d f r a c t io n s c l o s el y a k i n

t o t h e B E R N S T E I N - B O R E L r e s u l t , t h i s a g a i n b y i n t e r p r e t i n g t h e t h e o r e m i n t h e

s e t ti n g o f m e a s u r e t h e o r y .

T h e f ir st in o r d e r o f o c c u r r e n c e , FABER, r e g a r d e d t h e c o e x t e n s i o n o f BOREL's

d e n u m e r a b l e p r o b a b i l i ty a n d L EB ES GU E m e a s u r e a s a n o p e n q u e s t i o n w h i c h h e

w a s a t p a i n s t o r a i s e ; H A U S D O R F F w a s m u c h m o r e a s s e r t i v e a n d w e n t s o f a r a s t o

o f fe r e x p li c it l y a n " a r b i t r a r y " d e f i n i t io n o f p r o b a b i l i t y a s ( L EB E SG U E ) m e a s u r e .R A D E M A C H E R r e g a r d e d t h e d u a l v i e w p o i n t s a s s u f f i c ie n t l y w e l l a c c e p t e d a s to b e

a l m o s t w i t h o u t n e e d o f c o m m e n t . T h e w o r k o f F A BE R a n d R A D EM A C H ER (t h e

1 As w as the view point adopted by BOREL n 1905.

Axiomatic ProbabiIity 177

l a t t e r u n i n t e n t i o n a l l y d u p l i c a t i n g t h e f o r m e r ) w i l l b e d e a l t w i t h f i r s t . T o p r e s e r v e

c h r o n o l o g i c a l o r d e r w e i n t e r s p e r s e b e t w e e n t h e m H A U S D O R F F 'S b r i e f b u t h i s t o r i -

c a l l y s i g n i f i c a n t c o m m e n t s o n m e a s u r e a s a p o s s i b l e de f in i t i on o f p r o b a b i l i ty .

T h e s u b s e q u e n t s e c ti o n is a n e x p o s i t i o n o f m o r e t e c h n i c a l a s p e c t s o f

H A U S D O R F F 's w o r k , o n b o t h b i n a r y a n d c o n t i n u e d f r a c ti o n e x p a n s io n s .

9 .2 . T h e C o n t r i b u t i o n o f G . F a b e r : H i s Q u e r y o n t h e R e l a t i o n

o f P r o b a b i l it y t o M e a s u r e

G . F A B E R (1 9 1 0) a n d H . R A D EM A C H E R ( 19 1 8 ) e a c h c o n s t r u c t a c o n t i n u o u s

m o n o t o n e f u n c t i o n i n [ 0 , 1 ] w h o s e d i f fe r e n c e q u o t i e n t s a t a p o i n t x d e p e n d o n t h e

d i s t r i b u t i o n o f d i g it s in t h e e x p a n s i o n o f x . T h e i r c o n s t r u c t i o n s a r e v i r t u a l l y

i d e n ti c a l. I n e a c h c a s e a m o n o t o n e f u n c t i o n f i s c o n s t r u c t e d ( as th e l i m i t o f a n

a u x i l i a r y s e q u e n c e { fn }) s u c h t h a t t h e v a l u e o f t h e d i f f e re n c e q u o t i e n t

f i - ~ - f \ 1 0" /

d e p e n d s o n l y o n~ k ( n )- - , k = 0 , 1 , 2 . . . . . 9

w h e r e 7 k ( n ) c o u n t s t h e n u m b e r o f k ' s a m o n g t h e f ir s t n d ig i ts i n t h e d e c i m a l

e x p a n s i o n o f x , a n d w h e r e t h e i n t e g e r x n is s u c h t h a t

X n - - 1 Xn- - - < x < - -

10~ 10""

T h e m a i n r e s u l t w h i c h f o l l o w s f r o m f u r t h e r s p e c i fi c s o f t h e c o n s t r u c t i o n i s t h a t t h e

s e t o f p o i n t s x f o r w h i c h

l i r a 7k (n ) # 1. ~ n 10

a r e p o i n t s o f n o n - d i f f e r e n t ia b i l i ty . W e n o t e t h a t a s i m i l a r c o n s t r u c t i o n c a n b e

a p p l i e d t o b i n a r y e x p a n s i o n s , o r t o e x p a n s i o n s i n a n y g i v e n b a s e , a n d i n e v e r y c a s e

t h e r e s u l t i s t h a t t h e n u m b e r s w h i c h a r e n o t " n o r m a l " w i t h r e s p e c t t o t h e g i v e n

b a s i s , in t h e s e n s e o f B OR E L, a r e p o i n t s o f n o n - d i f f e r e n t i a b i l i t y .

B o t h F A B E R a n d R A D E M A C H E R t h e n a p p e a l e d t o t h e p o w e r f u l r e s u l t o f

L E BE S G UE w h i c h a s s e r t s t h a t f o r e v e r y m o n o t o n e f u n c t io n t h e s et o f p o i n t s o f n o n -

d i f f e r e n ti a b i l it y i s o f m e a s u r e z e r o .

T h u s , t h e f u n c t i o n s c o n s t r u c t e d b y F AB ER a n d RADEMACHER, t o g e t h e r w i t h

t h is t h e o r e m o f L EB E SG U E, i m p l y t h a t t h e s e t o f p o i n t s w h e r e f o r a t l e a s t o n e c h o i c e

o f k = 0 , 1 . . . . 9l im 7k(n) 1

n ~ o o n : : t = l O '

( t h a t is, th e s e t o f n u m b e r s n o n - n o r m a l t o t h e b a s e 1 0) i s o f m e a s u r e z e r o .

A c o m p a r i s o n b e t w e e n t h e t w o p a p e r s , F A B E R ' S d a t e d 1 9 1 0 a n d

RADEMACHER'S 1 9 1 8 , s h o w s t h e e x t e n t t o w h i c h t h e e q u i v a l e n c e b e t w e e n

L E B E S G U E m e a s u r e o n [ 0 , 1 ] a n d g e o m e t r i c p r o b a b i l i t y o n [ 0 , 1 ] h a d s h i f t e d i t s

s t a tu s f r o m c o n j e c t u r a l to a v i rt u a l ly u n a n i m o u s c o n v e n t i o n .

W h e n F AB ER , w h o s e m a i n a i m w a s in a s o m e w h a t d i f f er e n t d i r ec t io n , a c h i e v e d

a l m o s t i n a d v e r t e n t l y t h e re s u l t t h a t a l m o s t a ll n u m b e r s a r e n o r m a l ( th e " S t r o n g

L a w o f L a r g e N u m b e r s " ) , h e p a u s e d t o i n te r p o l a te s o m e c o m m e n t s ; t h e se s e rv e as a

p r i m a r y s o u r c e f o r e x a m i n i n g t h e e x t e n t t o w h i c h B O R E L ' s p a p e r l e ft i ts r e a d e r s

u n s u r e a s t o th e r e l a t io n b e t w e e n p r o b a b i l i t y t h e o r y a n d m e a s u r e t h e o r y 1:

T h e s e t o f p o i n t s f o r w h i c h l i m 7 "4 = 1 o r l i m ~ " = ~ 1, i s o f m e a s u r e z e r o .n~oo # . . . . # ,

T h i s t h e o r e m a p p e a r s i n t e re s t in g t o m e f r o m m a n y p o i n t s o f v ie w .

F i r s t i t g i v e s a s i m p l e e x a m p l e o f a s e t whi ch i s no t on l y eve rywhe re dense bu ta l so has t he card i na l i t y o f t he con t i nuum i n eve ry i n t e rva l , however sm a l l , and

none t he l e s s has m easure z e ro .

B o r e l r e c e n t l y p r o v e d , a f t e r f o r m u l a t i n g s u i t a b l e d e f in i t io n s c o n c e r n i n g

d e n u m e r a b l e p r o b a b i l it i e s , t h a t t h e p r o b a b i l i t y t h a t a p o i n t b e l o n g t o th e a b o v e

s e t is ze r o. T h e c o m p a r i s o n o f t h e a b o v e t h e o r e m w i t h B o r e l 's r e s u lt s u g g e st s t h e

q u e s t i o n :

I s t h e p r o b a b i l i t y - a c c o r d i n g t o t h e B o r e l s e t -u p w h i c h p o s s i b l y m i g h t n e e d

t o b e e x te n d e d t o a n s w e r th i s q u e s ti o n - t h a t a n u m b e r b e l o n g s t o a p r e s c r i b e d

s e t o f z e r o m e a s u r e , a l w a y s e q u a l t o z e r o ? A n d c o n v e r s e l y : is a se t a l w a y s o f

m e a s u r e z e r o , if t h e p r o b a b i l i t y t h a t a p o i n t b e l o n g s t o i t is e q u a l t o z e r o ? ( I ta l ic s

i n t he o r i g ina l . ) FABER (1910 : 400 ) .

T h e p a s s a g e j u s t c i t e d is e v i d e n c e t h a t , f o r F AB E R, t h e c l e a r - c u t i d e n t i f i c a ti o n o f

g e o m e t r i c p r o b a b i l i t y o n [ 0, 1 ] w i t h L EB ES GU E m e a s u r e h a d n o t q u i t e t a k e n p l a c e

b y 1 91 0. B O R E L 's o w n s u g g e s t i o n o f 19 05 t h a t s u c h g e o m e t r i c p r o b a b i l i t y b e

i d e n t if i e d w i t h L EB ES GU E m e a s u r e s e e m s t o h a v e b e e n u n k n o w n t o F A BE R. (R e c a l l

t h a t i n 1 9 09 t h i s s u g g e s ti o n , r e d u c e d t o a m e r e a s i d e , w a s d i s m i s s e d b y B O RE L

h i m s e l f . ) I t i s c l e a r t h a t F A B ER , o n t h e h e e l ' s o f B OR E L 'S p a p e r o f 1 90 9, w a s

g r a p p l i n g w i t h th e s a m e n a s c e n t i d e n t if i ca t io n , m o t i v a t e d b y th e " a c c i d e n t a l "

a l t e r n a t iv e a p p r o a c h h e f o u n d t o B O R E L 's S t r o n g L a w .A s q u o t a t i o n s f r o m B O RE L h a v e s h o w n , a l l t h a t w a s n e e d e d t o a n s w e r F A BE R 'S

q u e s t i o n a f f i r m a t i v e l y , a n d t o m a k e t h i s i d e n t i f i c a t i o n a b s o l u t e l y e x p l ic i t, w a s a

c a r e fu l r e - e x a m i n a t i o n o f B O R E L 's o w n p a p e r w i t h e m p h a s i s o n t h e d u a l i t y

b e t w e e n t h e " p o i n t d e v u e l o g i q u e " a n d t h e " p o i n t d e r u e g 6 o m 6 t r i q u e " .

Neit her FABER n o r BOREL c a n b e r e a l is t i c a ll y c h a r g e d w i t h o b t u s e n e s s ; BOREL

a c h i e v e d h is r e s u lt a s a n a p p l i c a t i o n o f a n a b s t r a c t t h e o r e m o n i n d e p e n d e n t t ri al s,

F AB ER b y c o n s i d e r i n g f u n c t i o n s o f a r e a l v a r i a b l e d e f i n e d o n [ 0 , 1] . T h e s i t u a t i o n

m a y b e d e s c r i b e d b y s a y i n g t h a t B OR EL , p e r f e r ri n g t h e " p o i n t d e v u e l o g iq u e ," w a s

e x p l o r i n g t h e p r o d u c t m e a s u r e a v a i l a b l e o n a c e r t a i n p r o d u c t s p a c e , a n d F A B E R

w a s e x p l o r i n g t h e c o n s i d e r a b l y m o r e f a m i l i ar t e rr i t o r y o f m e a s u r a b l e f u n c t io n sd e f i n e d o n [ 0 , 1 ] .

1 I n FABER'Snotation 7,(/~) is the num ber of l's (0%) n the first n terms of he binary exp ansion of hegiven real number.

T h e q u e s t i o n o f t h e m a p p i n g b e t w e e n t h e p r o d u c t s p a c e i m p l i c it in B OR EL a n d

t h e f a m i l i a r m e a s u r e s t r u c t u r e o f [0 , 1 ] w a s n o d o u b t o f l e s se r i n t e r e s t t o BO R EL a n d

F A B E R t h a n t h e i r p r i m a r y b u t d i s t i n c t a i m s . A l t h o u g h i t i s d i f f i c u l t f o r a

c o n t e m p o r a r y r e a d e r t o r e a d B O R EL 'S p a p e r o f 19 09 w i t h o u t e m p l o y i n g t h e

a d v a n t a g e s s u p p l i e d b y h i n d s i g h t , w e t a k e F A B E R ' s a b o v e r e m a r k a s f u r t h e r

e v i d e n c e c o n f i r m i n g o u r a s s e r t i o n t h a t B O RE L fa i le d t o s ee , a n d c e r t a i n l y t o s t a t e,

t h e i d e n t i fi c a t i o n i n 1 90 9. I n d e e d , w e t h i n k W I N T N E R w a s b e i n g s o m e w h a t o v e r -

g e n e r o u s i n h is a s s e s s m e n t o f B O RE L w h e n h e w r o t e :

H i s t o r ic a l ly , t h e w h o l e d e v e l o p m e n t w a s i n i ti a te d b y B o r e l' s f o r m u l a t i o n

a n d p r o o f o f h i s " e i t h e r 0 o r 1 " t h e o r e m ( R e n d . P a l e r m o , v o l . 2 9 (1 90 9), p p . 2 4 7

2 71 ). T o d a y i t i s e a s y , b u t a t t h a t t i m e it w a s a t r u e m a t h e m a t i c a l a c h i e v e m e n t ,

t o t h i n k o f t h e o r d i n a r y m e a s u r e o n t h e i n t e r v a l 0 < x < 1 a s a p r o d u c t m e a s u r e

i n a n i n f i n i t e p r o d u c t s p a c e , t h e f a c t o r s b e i n g t h e b i n a r y s p a c e s c o r r e s p o n d i n gt o t h e d y a d i c e x p a n s i o n o f x . W I N T N E R (1 9 4 1 : 1 8 2).

9 .3 . Radem acher and Ha usdo r f f The Evidence for t he Evo lu tion

o f a Po in t o f V iew

E i g h t y e a r s a f t e r t h e p u b l i c a t i o n o f F A B ER (1 91 0), R A D E M A CH E R , in i g n o r a n c e

o f F A B E R ' s r e s u lt , p r o v e d i t a n e w b y a v i r t u a l l y i d e n t ic a l c o m s t r u c t i o n o f a

m o n o t o n e f u n c t i o n w h o s e n o n - d i f f e r e n t i a b i l i t y w a s a s s u r e d a t t h e n o n - n o r m a l

n u m b e r s . I n t h e i n t e r im , H A U S D O R F F 'S Grundzi ige der M enge nlehre h a d a p p e a r ed .

T h i s v o l u m e h e lp e d d i s s e m i n a t e m a n y c o n c e p t s w h i c h w e r e b e c o m i n g p a r t o f t h em a t h e m a t i c a l " c u l t u r e " o f t h e t im e , f o r e x a m p l e s u c h c o n c e p t s a s fi el d s o f s e ts ,

B O R EL s e t s , l im i n f a n d l i ra s u p o f s e ts .

I n p a r t i c u l a r , H A U S D O R F F g a v e a b r i e f i n t r o d u c t i o n t o t h e L E B ES G UE t h e o r y o f

m e a s u r e a n d o f t h e L E B ES GU E i n t eg r a l . A s examples o f m e a s u r e t h e o r y H A U S -

D O R F F p r e s e n t e d b o t h t h e B O R EL r e su l t o n d e c i m a l s a n d a c l o se l y r e l a t e d

c o n t i n u e d f r a c t i o n r e s u l t , b u t w i t h p r o o f s i n n o w a y r e s t i n g o n B O R E L ' S

i n v e s t i g a t i o n o f 1 90 9 o f t h e p r o b a b i l i t y o f i n f in i te l y m a n y s u c c e s se s i n a

d e n u m e r a b l e s e q u e n c e o f t r ia l s o r i ts e x t e n s i o n i n 1 9 12 .

I n c o n t r a s t w i th t h e o r i g i n a l t r e a t m e n t o f B OR EL , b o t h t h e S t r o n g L a w a n d t h e

r e s u l t o n c o n t i n u e d f r a c t i o n s a r e t r e a t e d s o l e ly as e x a m p l e s o f t h e t h e o r y o fL E BE S GU E m e a s u r e . HAUSDORFF ' s t r e a t m e n t o f t h e s e t h e o r e m s w i ll b e g i v e n in t h e

n e x t s e c t i o n . W h a t i s n e e d e d h e r e , a s t h e n e c e s s a r y h i s t o r i c a l s e t t i n g o f

R A D E M A CH E R 'S p a p e r 1 9 18 , i s t h e p r e c i s e l a n g u a g e o f H A U S D O R F F o n t h e r e l a t i o n

b e t w e e n m e a s u r e t h e o r y a n d p r o b a b i l i s t i c t e r m i n o l o g y , a n d h i s r e s t a t e m e n t o f

B O R E L 's S t r o n g L a w . 1

W e r e m a r k t h a t m a n y t h e o r e m s c o n c e rn i n g th e m e a s u r e o f p o i n t- s et s

a p p e a r p e r h a p s m o r e i n tu i ti v el y , if o n e e x p r es s es t h e m i n t h e l a n g u a g e o f

p r o b a b i l i t y . I f t w o s e t s P a n d M a r e m e a s u r a b l e , a n d M i n p a r t i c u l a r i s o f

p o s i t iv e m e a s u r e , t h e n o n e c a n d e fi ne , b y m e a n s o f t h e q u o t i e n t f ( P ) / f ( M ) i f P

___ M , o r m or e ge ne ra l l y by f ( P c~ M ) / f ( M ) , t h e p r o b a b i l i t y t h a t a p o i n t o f Mb e l o n g s t o P . I f w e c o n s i d e r o n l y s u b s e t s o f a fi x ed s e t M , o f m e a s u r e 1, t h e n f ( P )

W e h av e taken the libe rty of altering HAUSDORFF'S original notation for u ni on and in-tersection, ~(P 1, P2) and ~(P 1, P2) respectively.

180 J. BARONE 8z; A. NOVIKOFF

= p i s t h e p r o b a b i l i t y t h a t a p o i n t b e l o n g s t o t h e s e t P . I f P = / ' 1 w P 2, P ' = / ° 1 c~ P 2,

t h e n p + p ' = p 1 + P 2 ; P i s t h e p r o b a b i l i t y t h a t a p o i n t b e l o n g s t o P 1 o r P2, P' i s th e

p r o b a b i l i t y t h a t a p o i n t b e l o n g s t o P1 a s w e l l as t o P 2. I f P1 a n d P2 h a v e n o p o i n t s

i n c o m m o n , t h e n p = p ~ + p 2 . F u r t h e r m o r e p ' - - p 1 ~ - ( if p i > 0 ) ; t h e p r o b a b i l i -

t y t h a t a p o i n t b e l o n g s s i m u l t a n e o u s l y t o P t a n d P2 i s t h e p r o b a b i l i t y t h a t i t

b e l o n g s t o P I m u l t i p l i e d b y t h e p r o b a b i l i t y t h a t a p o i n t o f Pa s h o u l d b e l o n g t o

P 2- T h e f o r m u l a f o r s o - c a ll e d " i n d e p e n d e n t " e v e n t s p' =p ~ P2 is o f c o u r s e n o t

v a l i d i n g e n e r a l. I t i s a l s o p e r f e c t l y c l e a r t h a t f r o m t h is ( b y a n d l a r g e a r b i t r a r y )

d e f i n it i o n , p r o b a b i l i t y 0 is n o t t h e e x p r e s s i o n o f i m p o s s i b i l i t y , a n d p r o b a b i l i t y 1

n o t t h a t o f c e r t a i n t y ; f o r 0 is t h e p r o b a b i l i t y t h a t a p o i n t b e l o n g s t o a s e t o f z e r o

m e a s u r e ( w h i c h m i g h t s ti ll b e o f t h e c a r d i n a l i t y o f t h e c o n t i n u u m ) . H A U S D O R FF

( 1 9 1 4 : 4 1 6 - 4 1 7 )

T h i s c i t a t i o n s h o u l d l e a v e n o d o u b t a s to h o w e a r l y FA B ER 'S q u e s t i o n o f 1 91 0

w a s a n s w e r e d , a n d h o w e a r l y a s c r u p u l o u s l y c l e ar e x p r e ss i o n o f p r o b a b i l i t y w a s

p u b l i s h e d - i n c lu d i n g i n d e p e n d e n c e , c o u n t a b l e a d d i ti v i ty , c o n d i t i o n a l p r o b a b i l i ty ,

a n d e v e n t h e n o t i o n o f s a m p l e s p a c e . H A U S D O R F F'S r e m a r k s , p r e f a t o r y f o r h i s

e x p l i c i t r e - w o r k i n g o f B O R E L'S r e s u l t s , a r e i n t h e s a m e v e i n a s B O R EL 'S p a p e r o f

1 9 0 5 , b u t t h e y r e a c h c o n s i d e r a b l y f u r t h e r .

T w o p a g e s l a t e r , H A U S D O R F F s t a t e s BOREL's S t r o n g L a w a s f o l l o w s :

I I . Se t s o f Dyad ic Dec imals . W e c o n s i d e r a n i r r a t i o n a l n u m b e r x b e t w e e n 0

a n d 1 a n d e x p a n d i t i n a d y a d i c d e c i m a l [ f o o t n o t e o m i t t e d ] :

x 1 x2 x 3x = ~ - + ~ + 3 X + . . . . (Xx , Xz , X3 . . . . ) ( x , = 0 , 1 ) .

A m o n g t h e f i r st n d i g it s w il l a p p e a r p z e r o s a n d q = n - p o n e s. T h e n o n e h a s t h e

T h e o r e m ( E. B o r e l) :

T h e s e t o f x f o r w h i c h l i m P = l_ h a s m e a s u r e 1.

O r : t h e c o m p l e m e n t , i .e ., t h e s e t o f t h o s e x , f o r w h i c h p- i s e i t h e r n o n -n

c o n v e r g e n t , o r d o e s n o t c o n v e r g e t o ½, h a s m e a s u r e 0 . T h e r e i s t h u s ap r o b a b i l i t y 1 t h a t t h e d y a d i c e x p a n s i o n o f x h a s a s y m p t o t i c a l l y as m a n y

z e r o s a s o n e s .

T h i s t h e o r e m is r e m a r k a b l e . O n t h e o n e h a n d i t s e e m s a p l a u s i b l e e x te n s i o n

o f t h e " L a w o f L a r g e N u m b e r s " t o t h e i n fi n it e ; o n t he o t h e r i t as s e rt s th e

e x i s t e n c e o f a l im i t o f a s e q u e n c e , a n d i n d e e d e v e n a p r e s c r i b e d v a l u e o f t h e l i m i t,

a v e r y sp e c i al c i rc u m s t a n c e , w h i c h o n e w o u l d h a v e h e l d a pr ior i t o b e

e x c e e d i n g l y u n l i k e l y . H A U S D O R F F ( 1 9 1 4 : 4 1 9 - 4 2 0 ) .

W e m a y t h u s s t a te t h a t f o r H A U SD O R F F t h e n o t i o n o f p r o b a b i l i t y is

e m p h a t i c a l l y s e e n a s a d e r i v e d o n e , r e s t i n g o n t h e m o r e f u n d a m e n t a l o n e o f

m e a s u r e . T h e s u b s e q u e n t p r o o f d o e s n o t r e f e r t o p r o b a b i l it i e s f o r n o t a t i o n ,

i n t u i ti o n o r m e t h o d , b u t e m p l o y e d s o le l y t h e l e n g t h s o f i n t e rv a l s o f c o v e r i n g s e ts

( se e b e lo w ) . T h e t h e o r e m s e r v e d H A U SD O R F F o n l y a s a n e x a m p l e , a l t h o u g h a v e r y

i n t e r e s t in g o n e , o f th e u s e o f c o u n t a b l e a d d i t i v i t y o f L EB ES G UE m e a s u r e .

T h i s is t h e b a c k g r o u n d o f RA D E M AC H ER 's p a p e r . T h e S t r o n g L a w c a n b e

v i e w e d a s a n a s s e r t i o n t h a t a c e r t a i n s e t h a s m e a s u r e 0 ; w h a t RADEMACHER

p r e s e n t e d i s a l s o i n th i s s e tt in g , b u t i n s t e a d o f a d i r e c t c a l c u l a t i o n o f s m a l l c o v e r i n g

s e t s , i n t h e m a n n e r o f HAUSDORFF, t h e p o w e r f u l L E B ES G UE t h e o r e m ( t h a t e v e r y

m o n o t o n e f u n c t i o n is a lm o s t e v e r y w h e r e d if f er e n ti a b le ) is e m p l o y e d . T h i s c o m p l e t e

a c c e p t a n c e o f t h e t w o - f o l d c h a r a c t e r o f t h e t h e o r e m , p r o b a b i l i st i c i n th e h a n d s o f

BOREL, m e a s u r e - t h e o r e t i c i n t h e h a n d s o f HAUSDORFF 1 is e v i d e n t in t h e w o r d i n g

o f R A D E M A C H E R ' s i n t r o d u c t i o n , w h i c h f o l l o w s .

O n e o w e s t h e f o l lo w i n g r e m a r k a b l e t h e o r e m t o M r . E . B o r e l [ f o o t n o t e

o m i t t e d ] :

D e n o t i n g b y n~ ( ? = 0 , 1 , 2 . . . . 9 ) t h e n u m b e r o f d ig i ts ? a m o n g t h e f i rs t n

p l a ce s in t h e d e c i m a l e x p a n s i o n o f a n u m b e r f r o m t h e i n t e r v a l b e t w e e n 0 a n d 1,

t h e n l i m n ~ _ 1n ~ n 10

e x c e p t f o r a n u l l- s e t Z , w h i c h h o w e v e r h a s t h e c a r d i n a l i t y o f t h e c o n t i n u u m

[ f o o t n o t e o m i t te d ] .

B o r e l e x p r e s s e s t h e t h e o r e m t h u s : t h e p r o b a b i l i t y t h a t , i n a d e c i m a l

e x p a n s i o n , t h e t e n d i g i t s d o n o t a s y m p t o t i c a l l y e q u a l l y o c c u r , i s z e r o , a n d

c a r r ie s o u t t h e p r o o f b y a n e x t e n s i o n o f B e r n o u l l i 's t h e o r e m o n p r o b a b i l i ty . A

d i r e c t p r o o f o f t h e t h e o r e m i n o u r f o r m u l a t i o n w a s g i v e n b y H a u s d o r f f , i n w h i c h

h e g a v e a n u p p e r e s t i m a t e o f t h e s e t Z b y m e a n s o f c o v e r i n g s w i t h i n t e rv a l s. T h e

f o l l o w i n g p r o o f i s p e r h a p s n o t s u p e r f l u o u s . . . R A D E M A C H E R ( 1 9 1 8 : 3 06 ).

T o u s , RADEMACHER's a s s e r t io n t h a t B O RE L c a r r i e d o u t t h e p r o o f b y " a n

e x t e n s i o n o f B E RN O U LL I's t h e o r e m " s e e m s r a t h e r m i s le a d i n g . I n d e e d , t h e t h e o r e m

is n o t a c o n s e q u e n c e o f t h e Z e r o - O n e L a w o f 19 09 , a n d t h e g a p is n o t r e a d i l y f il le d

( e v e n b y t h e e x t e n s i o n o f i t i n 1 9 12 ), w i t h o u t r e c o u r s e t o c o u n t a b l e s u b - a d d i t i v i t y i n

the "CANTELLI" m a n n e r a s w a s d o n e b y HAUSDORFF.

A s w e h a v e s e e n , e v e n b y 1 91 2, t h e d a t e o f h i s r e p l y t o B E R N ST E IN , B O R E L d o e s

n o t s e e m t o h a v e b e e n a d v i s e d b y LEBESGUE, B E R N ST E IN , o r a n y o n e e l se t h a t h i s

S t r o n g L a w o f L a r g e N u m b e r s d o e s n o t f o l lo w f r o m h is Z e r o - O n e L a w o f 19 09 .

W r i t in g i n 1 91 2, w h e n h e e v o l v e d a m o r e g e n e r a l Z e r o - O n e L a w t o c o v e r th e r e s u l t

o n c o n t i n u e d f r a c ti o n s , B O R EL m a d e n o a t t e m p t t o l e g i t im a t e h i s S t r o n g L a w

s i m i la r l y . F A BE R 's p r o o f i n 1 9 10 , in c o n t e s t a b l e a s it w a s, m u s t h a v e s e e m e d a n

" a c c i d e n t " , s i n c e i t b o r e n o e v i d e n t r e l a t i o n t o p r o b a b i l i t y ( c f FA BER's ow n

p u z z l e m e n t r e m a r k e d a b o v e ) .

T h i s b r i n g s t h e n a r r a t i v e u p t o HAUSDORFF, w h o i n 1 9 1 4 s i m u l t a n e o u s l y

a c c o m p l i s h e d s e v e r al r e m a r k a b l e t h i n g s : h e c o n s id e r s b o t h t h e d e c i m a l a n d

c o n t i n u e d f r a c t i o n c a se in t u rn , r e c a s t in g t h e m b o t h a s t h e o r e m s i n m e a s u r e t h e o r y .

H e p r o v e s t h e d e c i m a l c a s e b y s u b - a d d i t i v i t y ( i . e . , i n t h e " C A N T E L L I " m a n n e r ) ,

i n t r o d u c i n g a p r o w e r f u l d e v i c e t o e v a d e a n a p p e a l t o t h e C e n t r a l L i m i t T h e o r e m .

H e a l s o p r o v e s a c o n t i n u e d f r a c t i o n r es u l t a k i n t o t h e BERNSTEIN-BOREL one , a s

c l e a r l y a s d i d BERNSTEIN, a l t h o u g h w i t h o u t i n t r o d u c i n g t h e m o d e r n l a n g u a g e o f

c o n d i t i o n a l p r o b a b i l it i es .

1 And also FABER, although unknown at the time to RADEMACHERand cited by HAUSDORFF.

His p roo fs are a l l in the langu age of (LEBESGUE) mea sure , w i th m od er n

not at ion s , such as l im sup E, for a sequ ence {En} of se ts . As to the r e la t io n o f his

(measure - theore t i c ) a s se r t ions to those of BOREL, which were couched in the

language of probabi l i ty , HAUSDORFF expl i c i tly d isposes of th is in the pre f a toryr e ma ks c i te d a bove .

10. Hausdorff's "Grundziige der Mengenlehre":

A Notable Advance in Technique

10.1. General Background

This sect ion wi l l c i te vi r tual ly a l l the probabi l is t ic references to be found in

HAUSDORFF's Grundzfige der Mengenlehre. The f i r s t p robabi l i s t i c re fe rence ,

a l r e a dy c i te d , c onc e r ns t he r e l a t i on be t w e e n t he voc a bu l a r y o f p r oba b i l i t y a nd t ha t

of me asur e the ory. In this sect ion we discuss HAUSDORFF's t re atm en t of dy adic

e xpa ns i ons a nd c on t i nue d f r a c t i ons v i a me a s u r e t he o r y .

C ha p t e r X o f t he Grundziige (1914) is the locu s o f a l l of the abo ve i tems. T his

chap te r i s t i t led "C on ten t o f Poin t -Se t s ." I ts f i rs t pa r agr aph begins wi th genera l i t ie s

concerning length, measure and area in EUCLIDEAN space, and a his tor ical sketch.

Th e olde r the or y of con ten t asso cia ted wi th CANTOR, HANKEL, PEANO, and

JORDAN, conc ern ed i t se l f wi th f ini te add i t ivi ty fo r dis joint se ts ; the newe r theo ry,

due to BOREL and LEBESGUE, specif ical ly ad ded co un tab le ad di t ivi ty. HAUS-

DORFF ci ted LEBESGUE'S pr ob lem and for m ula ted i t thus :

. .. L e be s gue f o r m u l a t e d t he p r ob l e m o f a s s oci a ti ng t o e ve r y bou nde d s et A in n -d i me ns i ona l s pa c e E , , a s " c on t e n t , " a num be r f (A)> 0 sat is fying the fol lowing

c ond i t i ons :

(a ) Congruent s e t s have the same measure .

( f i ) The uni t cube has conten t 1 .

(7) f( A + B) = f( A) + f(B).(6) f (A + B + C +...) = f(A) + f(B ) + f( C) +... f o r a boun de d s um o f c oun -

tably many sets . HAUSDORFF (1914: 401).

HAUSDORFF imm edia te ly gave an exam ple show ing the im poss ib i l i ty of th i s

problem in fu l l genera l i ty . The example , s t i l l t he mos t popula r one (cf ROYDEN

(1965: 52-55) , map s the l ine on to th e c i rc le ; on the c i rc le i t exhibi ts a se t which is

d i s jo in t f rom a l l i t s images under ra t iona l ro ta t ions . Fur the r , th i s s e t and i t s

(necessa r i ly congruent ) images under a l l ra t iona l ro ta t ions i s a d i s jo in t decom-

pos i t ion of the en t i re c i rc le .

I t is of interes t to n ote th at this ex am ple is no t a t t r ibu ted by HAUSDORFF to a ny

a u t ho r . H ow e ve r , i n t he r e f e r enc e s ne a r t he e nd o f t he boo k t w o s ou r c es o f exa mpl e s

of non -m easu rabl e se ts a re g iven: G . VITALI 's Sul ProbIema della Misura dei Gruppi

di Punti di una Retta, a nd A . SCHOENFLIES'Mengenlehre. T he l a t t e r c on t a i n s

severa l examples of non -me asur able set s, inc luding the o ne g iven by HAUSDORFF.

Th e text of SCHOENFLIES (SCHOENFLIES (1913: 377, especia l ly fo otn ote 2))

indicates interes t ingly that this s imple example was in fact due to HAUSDORFF

himse l f , and communica ted d i rec t ly .

HAUSDORFF fur the r com me nte d on the im poss ib i l i ty o f the " easy LEBESGUE

problem" (employing the t e rminology of NATANSON (1961: 80) ) in which

r e q u i r e m e n t 6 is d r o p p e d i n sp a c e s o f t h r e e o r m o r e d i m e n s i o n s . T h e e x a m p l e

w h i c h e s t a b l i s h e s t h i s is g i v e n i n a n a p p e n d i x ; i t a l s o i s d u e t o H A U S D O R F F, w h o

a g a i n o m i t s a n y c l a i m o f a u t h o r s h i p o r p r io r i t y .

T h e s e c o n d p a r a g r a p h o f H A U SD O R FF ( 19 14 ), C h a p t e r X , is c o n c e r n e d w i t h t h e

t h e o r y o f P EA N O -JO R D AN c o n t e n t (c f H A W K I N S ( 1 9 7 0 : 8 6 - 9 6 ) f o r a h i s t o r y o f

t h e s e d e v e l o p m e n t s ).

P a r a g r a p h t h r e e , a b o u t n i n e p a g e s i n l e n g t h , i s a c o n c i s e a n d c o m p l e t e

t r e a t m e n t o f th e t h e o r y o f L EB E SG U E m e a s u r e i n t h e p l an e . C o u n t a b l e a d d i t i v i ty i n

a l l o f i ts f o r m s i s e s t a b l is h e d . C o u n t a b l e s u b - a d d i t i v i t y i s r i g o r o u s l y d e d u c e d

( HAUSDORFF ( 1914 : 414 : f o r m ul a ( 6 ) ) .

T h e n e x t, o r f o u r t h p a r a g r a p h o f H A U SD O R FF 's C h a p t e r X , is d e v o t e d t o

a p p l i c a t i o n s a n d e x a m p l e s . F o u r t o p i c s a r e d i s c u s s e d . T h e f i r s t c o n c e r n s n o n -

m e a s u r a b l e s et s, th e f o u r t h c o n c e r n s s e q u e n c e s o f m e a s u r a b l e f u n c t i o n s a n d t h e i r

c o n v e r g e n c e p r o p e r t i e s , su c h a s a l m o s t u n i f o r m c o n v e r g e n c e . It is to p i c s t w o a n d

t h r e e t h a t r e p r e s e n t HAUSDORFF's d i r e c t c o n t r i b u t i o n t o t h e d e v e l o p m e n t o f t h e

i d e a s c o n t a i n e d i n B O R E L's p a p e r o f 1 9 09 . I t is n o t e w o r t h y t h a t i n a b o o k o f 4 7 3

p a g e s , o n l y n i n e o f w h i c h a r e d e v o t e d t o t h e t h e o r y o f m e a s u r e , t w o o f t h e f o u r

t o p i c s c h o s e n t o i l l u s t ra t e m e a s u r e t h e o r y s t e m f r o m B O R E L'S p a p e r o f 1 9 09 . I t is

e s p e c i al l y t o b e n o t e d t h a t H A U SD O R FF p r e s e n t e d t h e s e t w o t o p ic s , t h e S t r o n g L a w

a n d c o n t i n u e d f r ac t io n s , in a c o m p l e t e l y o r ig i n a l f r a m e w o r k , i n w h i c h h e o f f e r e d

i n c o n t e s t a b l e p r o o f s . H A U SD O R FF t h e r e b y e m p h a s i z e d , t h e c l a r it y a n d s c o p e o f t h e

m e t h o d s o f m e a s u r e t h e o r y o n c e t h e f u n d a m e n t a l p r o p e r t ie s o f m e a s u r e h a v e b e e n

c l e a r l y l a i d d o w n .

10.2 . Haus dorf f s P roo f o f the S trong Law :

Ihe Use of M om en ts and BienaymO-Tchebycheff Type Inequalities

HAUSDORFF's f o r m u l a t i o n o f th e BOREL S t r o n g L a w h a s b e e n c i te d a b o v e . W e

n o w s k e t c h H A U SD O R FF 'S p r o o f , d i ff e ri n g o n l y i n m i n o r n o t a t i o n a l m o d i f i c a t io n s

f r o m t h e o r i g i n a l .

T o b e g i n w i t h , th e s e t o f a ll ir r a t i o n a l x w h o s e d y a d i c e x p a n s i o n s b e g i n w i th t h e

b i n a r y d i g i ts b l , b 2 . . . , b n, c o n s is t s o f t h e i r r a t i o n a l n u m b e r s i n t h e i n t e r v a l

bl b2 bn b l b 2 b n + l~ + ~ + - . . + ~ < x < ~ + ~ + . . . + 2 ~

a n d is t h e r e f o r e o f m e a s u r e 1 /2 n. ( T h e o m i s s i o n o f r a t i o n a l n u m b e r s , a s e t o f m e a s u r e

z e r o , p e r m i t s t h i s i d e n t i f i c a t i o n o f m e a s u r e w i t h l e n g th . ) T h e s e t o f t h o s e ( i r r a ti o n a l )

x w h i c h h a v e p r e c i s e l y k z e r o s a n d n - k o n e s in th e f i rs t n t e r m s c o n s i s ts o f ( ~ )

p a i r w i s e d i s j o i n t " i n t e r v a l s , " a l l o f m e a s u r e 1 / 2 n, a n d s o i s o f t o t a l m e a s u r e

F u r t h e r , l et e b e a p o s i t iv e n u m b e r a n d l e t E , ( e) b e t h e s e t o f t h o s e x f o r w h i c h

k 1 > ~~-~= .

T h e m e a s u r e o f th i s s e t, d e n o t e d p ,( e) , is t h e n g i v e n b y

1p , ( e ) ( ~ ) 2

w h e r e t h e s u m ~ is t a k e n o v e r t h o s e v a l u e s o f k fo r w h i c h

~_ 1 ~ e .

A t t h i s p o i n t I- IA U S D OR F F i n t r o d u c e s a t e c h n i c a l d e v i c e d e s t i n e d t o i n f l u e n c e

succes so rs ( such as STEINHAUS) w h i c h i n v o l v es e s t i m a t i n g e v e n - o rd e r m o m e n t s o f

t h e r a n d o m v a r ia b l e n - 2 " B y m e a n s o f t h is d e v ic e ( c f § 10 .3 be low) HAUSDORFF

s u c c e e d e d in s h o w i n g t h a t

( k 1 ] 4 ( ~ ) 1 3 1

k= 0 \ n - 2 1 ~ < 1 6 n 2" (1 0.1 )

I n t e r m s o f t h e n o t a t i o n v n( x f o r t h e n u m b e r o f z e r o s b l , . . . , b , i n t h em o n g

d y a d i c e x p a n s i o n o f x, th e s u m a p p e a r i n g o n t h e l e f t- h a n d s id e o f t h e a b o v e

i n e q u a l i t y w o u l d n o w b e c a l l e d t h e 4 th m o m e n t o f t h e r a n d o m v a r i a b le

I t f o l l o w s t h a t

* ( ~ ) 1 * [ k 1 ] 4 ( ~ ) 1 3 1

e 4 Z ~ < ~ \ n 2 ] ~ < 1 6 n 2

P"(e) < 16 e4 n 2,

so th a t ~ p ,(e) conv e rges fo r eve ry ~ > 0 .1

T h e s t r o n g L a w n o w f o l lo w s w i t h r e m a r k a b l e r a p i d i t y 1. I t m u s t b e e m p h a s i z e d

t h a t t h e i n e q u a l i t y ( 10 .1 ) p l a y s t h e r o l e f o r H A U S D O RF F w h i c h e s t i m a t e s f r o m t h e

C e n t r a l L i m i t T h e o r e m h a d p l a y e d fo r B OR EL .

Le t E(~) = l im sup E,(e) = ~ U E,(~) .T h e n E (e ) is e x a c t l y t h e s e t o f i r r a t i o n a l xn= 1 n= N

i n t h e u n i t i n t e r v a l f o r w h i c h

v,(x) 1m 21 Thi s inequa lit y is of the sort probabil is ts call TCHEBYCHEFF (or BIENAYMI~-TCHEBYCHEFF) ype,

th dalthou gh based on 4 power s rather tha n 2 n powers. It lies at the heart ofth e brilliantly briefpro ofgiv en

by KAC (1959: 16-17).

Fo ur th power est imates were also used by F. CANTELLI n his proof(CANTELLI (1917 a) and (1917b)).

L e t A ~ ( e ) = P(E(e)) = m e a s u r e o f E (~ ). T h e s e ts ~ ) E , ( e) d e c r e a s e w i t h i n c r e a s i n gn = N

N . O b v i o u s l y

E(e) c ~) E,(e) for N = 1 , 2 , 3 , . . . .n ~ N

F o r e a c h f i x e d i n t e g e r N w e h a v e , b y m o n o t o n i c i t y a n d c o u n t a b l e s u b - a d d i t i v i t y ,

t h a t

P(E,(e)) = p,(~).~(e)<=p E,(e) <=

H a v i n g sh o w n th a t ~ pn(e ) co n v er g es fo r ev e r y e > 0 , t h e m e as u re E (e ) = A oo(e) i s 0

f o r e v e r y e > 0 ) , =N

I n p a r t i c u l a r ~ = ~ ) E ( ~ ) i s a c o u n t a b l e u n i o n o f s e ts o f m e a s u r e 0 , a n d b y

co u n ta b le ad d i t iv i ty , i s i t s e l f o f m e as u r e 0 . B u t l=~)1E (~) i s t h e se t o f x fo r w h ich

l i m n 2 > 0 . T h u s i f ~ p , ( e ) c o n v e r g e s f o r a l l e > 0 , th e r e i s p r o b a b i l i t y (i.e.,n~oo

m e a s u r e ) 1 t h a t :

= 0limbo n 2

o r e q u i v a l e n t l y

v,(x) _1_J i m n 2 = 0 .

A t t h e v e r y l e a s t , H A U S D O R F F g a v e t h e f i r s t (c f p r e c e d i n g f o o t n o t e )

" C A N T E L L I " - t y p e p r o o f t h a t ~ p , < o o i m p l i e s A o o = 0 . I t s e e m s c l e a r t h a t

H A U S D O R F F , n o t C A NT E LL I, s h o u l d b e g i v e n c r e d i t f o r t h i s l i n e o f r e a s o n i n g . N o t

o n l y d i d H A U S D O R F F a n t i c i p a t e C A N T E LL I b y t h r e e y e a r s , h e a l s o g i v e s a t r e a t m e n t

1 This same argument shows (suppressing the e), that if ~p ,< o o , where p,=P(E,), thenP(lim sup E,) = A ~ = 0 thanks to sub additivity and with no appea l to independence. In other words, this

is CANTELLI'S argument, but resting on the firm base of measure theory. CANTELLIhad rested thecorresponding argument on the ad hoe assumption that sub-additivity("BooLn'S inequality") could be

extended from the finite to the countably infinite case.As early as 1906, in his thesis, FRECHET prov ed a theorem to which th e "CANTELLI" esult is an

imm ediate corollary, but couched in the language of measure (cf FRgCHET 1906: 16)). If for each integern, E, is a measurable subset of the unit interval whose measure re(El)equals 1 -m l , then FR£CHET

°bserved that m(~]lE't>l-(m~+m2+'")"~ / n It is immediate (although it seems to have

passed

unrem arked in the literature) that i f ~ m, converges (the only case of interest) then m E, > 1 - e forn=

N sufficiently large, depending on e. This yields m(lim inf E,) = 1 at once w ithout the intervention of any"independence" assumption. Since m,=m(E~,), his can be rephrased as follows: ~m(U,) convergesimplies re(lira inf U, )= 0, precisely th e "CANTELLI"result.

w h i c h i s i n o n e r e s p e c t s u p e r i o r : t h e s u b - a d d i t i v i t y

kn=N / n=N

i s s een a s a consequence o f t h e g e n e r a l p r o p e r t i e s o f m e a s u r e , n a m e l y , n o n -

n e g a t i v i t y a n d c o u n t a b l e a d d i t iv i ty , a n d a d o m a i n o f d e f in i ti o n c o n s is t in g o f a a -

a l g e b r a o f se ts . B y c o n t r a s t , C A N T E L L I a s s e r t e d

/I n=N

w i t h u n s p e c i fi e d g e n e r a li ty , r e a s o n i n g b y p u r e a n a l o g y w i t h t h e c o r r e s p o n d i n g

i n e q u a l i t y w h e n n h a s a f in i te r a n g e . T h u s C A N T E L L I a s s u m e s t h a t t h e d e g r e e o f

g e n e r a l i t y i s n o t l i m i t e d t o g e o m e t r i c p r o b a b i l i t y ( t h e o n l y c a s e t h e n k n o w n f o r

w h i c h m e a s u r e t h e o r y w a s a p p l ic a b le ) . O n t h e o t h e r h a n d , h e i g n o r e s th e f a c t t h a t

c o u n t a b l e s u b - a d d i t i v i ty r e s ts o n t h e p r i o r a p p a r a t u s o f c o u n t a b l e a d d i t i v i t y a n d o--

a l g e b r a s . H A U S D O R F F ' S pro of seems to be the f irs t fu l ly accurate re-working of

BOREL'S "p r o o f " o f t h e BOREL Law o f Large Numbers .

H A U S D O R FF o b s e r v e d a l s o t h a t t h a n k s t o h is m e t h o d o f m o m e n t s , u s i n g (e v en )

m o m e n t s h i g h e r t h a n t h e f o u r t h , t h e re s u l t o f B O RE L c a n i n fa c t b e c o n s i d e r a b l y

s t r e n g t h e n e d : w h e r e B O RE L s h o w e d t h a t t h e s e t o f x f o r w h i c h

l : ol i m \ n 2 ]n ~ o o

i s 1 , H A U S D O R F F 'S m e t h o d e s t a b l i s h e d t h a t t h e s e t o f x f o r w h i c h

i s 1, f o r a n y 0 l e s s t h a n ½ . H e t h u s i n i t i a t e d a s e q u e n c e o f r e f i n e m e n t s o f B O R E L ' s

S t r o n g L a w e s t a b l is h i n g m o r e a n d m o r e p r e c i se i n f o r m a t i o n a s t o th e rate a t w h i c h

v , ( x ) a p p r o a c h e s ½ w i t h p r o b a b i l i t y 1.n

10.3 . H ausdo rf f ' s Meth od for Calculat ing M om ents

S u m m a r i z i n g t h e m a i n r e s u l t : H A U S D O R F F s u c c e e d e d i n e s t a b l i s h i n g t h e

B OR EL L a w o f L a r g e N u m b e r s w i t h o u t a p p e a l t o t h e C e n t r a l L i m i t T h e o r e m . T h e

c o r r e s p o n d i n g t o o l f o r H A U S D O R F F w a s t h e i n e q u a l i t y : 1

3 (10 ,k =0 \ n 2 1 ~ < ~ n 2"

1 The left-hand side of (10.2) wou ld now be denoted

where Xj are independent random variables eac h with probability } of havin g the value 0 andprobab ility ½of having the v alue 1.

H A U S D OR F F s h o w e d g e n e r a ll y h o w t o e x p re s s " m o m e n t s " s u c h a s

k = o ( 2 k - - n ) l 2

a s p o l y n o m i a l s i n n w i t h i n t e g e r c o e f f i c i e n t s . T h e e v e n v a l u e s o f l a r e t h e m o s t

s i g n if i ca n t , as in t h e c a s e l = 4 c it e d a b o v e . T h e s e p o l y n o m i a l s a r e e a s i l y b o u n d e d

a b o v e b y p u r e p o w e r s o f n w i t h a s l ig h t ly l a r g e r l e a d i n g c o e ff ic i en t . T h u s

H A U S D O R F F s h o w s h o w t o o b t a i n e s t i m a t e s s u c h a s

a n d h e n c e

A2mk=O

¢ 1k~0

k l >I f 2 ., m e a n s s u m o v e r t h o s e v a l u e s o f k s a t i s f y i n g = e t h e r e r e s u l t s f i n a l ly

e 2 , , ~ 1 < A 2 ~ 12 n n m 22m •

I n s h o w i ng h o w t he s e m o m e n t s c o u l d b e e s ti m a t e d a n d h o w a d v a n t a g e o u s t h ey

w e r e, H A U SD O R F F a c h ie v e d a n o t a b l e a d v a n c e i n m e t h o d . T h e a b i li ty t o m a k e s u c he s t im a t e s w a s d e s t i n e d t o p l a y a r o le i n l a t e r d e v e l o p m e n t s b o t h w h e n l = 4 a n d

w h e n l = 2 m f o r l a r g e m . I n d e e d , C A N T E L L I w a s t o m a k e u s e , t h r e e y e a r s l a t e r , o f

s i m i l a r t e c h n i q u e s i n v o l v i n g t h e 4 t h m o m e n t , i . e . , l = 4 , a n d S TE IN H A US c o n s i d -

e r e d l = 2 m f o r l a r g e v a l u e s o f m i n 1 92 3.

T o c o n c l u d e t h e s e r e m a r k s , w e s k e tc h H A U S D O R FF 'S m e t h o d o f o b t a i n i n g e x a c t

p o l y n o m i a l e x p r e s s i o n s f o r

HAUSDORFF f i r s t c o n s i d e r e d

a n d i n t r o d u c e d t h e c h a n g e o f v a r i a b l e s u = x + y , v = x - y . T h e n t h e d i f f e r e n t i a l

0o p e ra to r D = x ~xx - y ffyy sa t i s f ies

D ( f + g ) = D f + D g ,

D ( f g ) = ( O f ) g + f ( D g ),

D ( f ' ) = n f ~ - 1 (D f ) ,

D ( x k y . - k) = (2 k - n ) x k y" - k .

I n p a r t i c u l a r , D ( u ) = v , D ( v ) = u .

1 8 8 J . B A R O N E 8 ,: A . N O V I K O F F

T h e c a l c u l a t io n o f D2m(un) t h e n p r e s e n t s n o d i f fi cu l ty , a n d t h e n u m e r i c a l c h o i c e

x = y = ½ , w h i c h c o r r e s p o n d s t o u = 1 , v = 0 , r e s u l t s i n

i (2k-nl. . . . . 1 = \ k ] 2 " 'v = 0 k = O

an m th d e g r e e p o l y n o m i a l i n n.

1 0 . 4 . H a u s d o r f f ' s C o n t i n u e d F r a c t i o n T h e o r e m

L e t u s i n t r o d u c e o n c e m o r e t h e n o t a t i o n [ a 1 = m l , a 2 = m 2 . . . . . a , = m , ] t o s t a n d

f o r t h e s e t o f c o n t i n u e d f r a c t i o n s w h o s e f ir s t n e l e m e n t s a l , . . . , a , a r e t h e p r e s c r i b e d

i n t e g e r s m l . . . . , m , . T h e s e f o r m a n i n t e rv a l , t h e l e n g t h o f w h i c h w e d e n o t e b y

P [ a l = m l , . . . , a n = m , ] .

T h e s e t d e f i n e d b y

[ a l = m l , . . . , a , _ l = m , _ l , k < = a , < m ]

i s a l so a n i n t e r v a l ( b e i n g a u n i o n o f a d j a c e n t i n t e r v a ls o f t h e a b o v e t y p e ) , th e l e n g t h

is d e n o t e d b y

P [ a 1 = m 1 . . . , a , _ 1 = m , _ 1 , k < a , < m ] .

T h e r a t i o o f t h e a b o v e t o t h e l e n g t h P [ a l = m l , . . . , a , _ 1 = l ~ n - - 1] c a n b e d e n o t e d b y

P [ k < a , < m [ a l = m l , . . . , a ~ _ l = m ~ _ l ]

u s i n g t h e m o d e r n n o t a t i o n f o r c o n d i t i o n a l p r o b a b i l i t y a l t h o u g h t h i s n o t a t i o n i s n o ts t r i c t ly n e c e s s a r y ( a n d i n d e e d is n o t u s e d b y HAUSDORFF). H A U S D O R F F 'S p o i n t o f

d e p a r t u r e is a n i n e q u a l i t y

p ( k , m ) < P [ k < a , < m l a I = m I . . . . , a , _ 1 = m , _ 1] < ~ r ( k , m )

w h e r e p a n d a c a n b e c a l c u l a t e d e x p l ic i tl y , a n d d o n o t d e p e n d o n n , n o r o n

m l , . . . , m , _ 1 . H A U S D O R F F c a l c u l a t e s " b e s t p o s s i b l e " e x p r e s s i o n s f o r p a n d a ,

i n c l u d i n g t h e i r s o m e w h a t a l t e r e d f o r m i f k = 1 o r m = o e ( b u t n o t b o t h ) . I t f o l l o w s

r e a d i l y t h a t

p ( k , , m , ,) < P [ k , < = a , , < m , , [ k 1 < =a 1 < = m I . . . . , k n - 1 < = a n - l < = m , - 1] ~ O'(kn, rn,)

b y p u r e l y a l g e b ra i c m a n i p u l a t i o n s . W e h a v e d e s c r i b e d t h i s m a n i p u l a t i o n i n

d i s c u s s i n g B E R N S T E I N 's w o r k ; i t i s e s s e n t i a l l y i d e n t i c a l i n H A U S D O R F F 'S .

B y m u l t i p l i c a t i o n o f t h e s e i n e q u a l i t i e s f o r c o n s e c u t i v e v a l u e s o f n ( w h i c h c a n b e

i n t e r p r e t e d a s t h e C h a i n L a w o f P r o b a b i l i t y ) i t f o l lo w s t h a t

P l . .. P , < P [ k l < a l < m a , . . . , k n < a , < m , ] < 1 1 . . . ~r,

w h e r e w e h a v e i n t r o d u c e d

p j = p ( k ~ , m j ) , a j = a ( k ~ , m j )

f o r b r e v i t y .

B y c o u n t a b l e a d d i t i v i t y ( e x p l ic i t l y s o s ta t e d )

] ~ p j < = P [ k j < =aj < = m j , j = l , 2 , 3 . . . . ] ~ ~ I a j .1 1

H A U S D O R F F is i n t e r e s te d o n l y in c o n d i t i o n s o n t h e t w o n u m e r i c a l s e q u e n c e s { k ,}

a n d { m , } t h a t a s s u r e p o s i t i v e m e a s u r e t o t h e s e t [ k j < a j < m j , j = 1 , 2 , 3 , . . . ] . F o r t h i soo oo

i t i s c l e a r l y n e c e s s a r y t h a t I ~ a j > 0 a n d s u f f ic i e n t t h a t H p j > 0 . I n v i e w o f t h e s p e c i f i c1 1

f o r m o f p ( k , m ) , a ( k , m ) i t r e s u l t s t h a t p o s i t i v e m e a s u r e i s o b t a i n e d i f a n d o n l y i f k j = 1

e x c e p t f o r f i n i t e l y m a n y v a l u e s o f j a n d

1 c o n v e r g e s .j = l m r

H A U S D O R F F d i d n o t p u r s u e h i s c a l c u l a t i o n s f u r th e r , a s h a d B E R N S T E IN a n d

B O R E L , to o b s e r v e t h a t t h e s e t d e f i n e d b y [ k n < a n < m n f o r a ll b u t f i n it e l y m a n y

v a l u e s o f n = 1, 2 , 3 , . . . ] i s n e c e s s a r i l y o f m e a s u r e 1 w h e n i t i s p o s i t i v e ( a l t h o u g h t h e

r e s u l t is i m m e d i a t e u s i n g h i s t e c h n i q u e s ) . R e c a l l , i t w a s th i s r e s u l t w h i c h s oa p p e a l e d t o B O R E L t h a t h e h a d p o i n t e d i t o u t a s th e m o s t i n t e r e s t i n g i n h is e n t i re

p a p e r o f 1 90 9.

I n s u m m a r y , H A U S D O R F F u s e d c o n t i n u e d f r a c t i on s a s a n e x a m p l e o f t h e p o w e r

o f L E B ES G U E m e a s u r e , e s p e c i a l l y i ts c o u n t a b l e a d d i t i v i ty a n d w i d e d o m a i n o f

d e f i n i t io n . H e o b t a i n e d a r e s u l t a l o n g t h e l i n e s o f t h e B E R N S T E I N -B O R E L o n e . L i k e

B E R N S T EIN h e d id n o t m i s t a k e n l y a s s u m e o r e m p l o y i n d e p e n d e n c e , b u t r a t h e r

m a n i p u l a t e d t h e p r o b a b i l i t ie s o f m u t u a l l y d e p e n d e n t e v e n t s in a f u ll y a c c u r a t e

m a n n e r . L i k e B E R N ST E IN , H A U S D O R F F o b t a i n e d t h e p r o b a b i l i t i e s o f n o n - c y l i n d e r

s et s b y v a l id l im i t o p e r a t i o n s o n t h e p r o b a b i l i t y o f a p p r o x i m a t i n g c y l i n d e r s et s.

N o n e o f B O R E L ' s e a r l y s u c c e s s o r s f o l l o w e d h i s l e a d t o w a r d s t h e r i g o r o u sf u n d a t i o n o f a t h e o r y o f p r o b a b i l i t y c o n c e r n e d w i t h r e p e a t e d t r ia l s ( i n d e p e n d e n t o r

n o t) . N o t u n t i l s o m e o n e w h o p o s s e s s e d f ac i li ty w i t h a x i o m a t i c a l l y b a s e d m e a s u r e

t h e o r y a n d w h o s h a r e d h i s p r i m a r y c o n c e r n w i th p r o b a b i l i t y ( in p a r t i c u l a r w i t h

r e p e a t e d t r i a ls ) w a s B O R E L t o h a v e a s u c c e s s o r a s i m p o r t a n t a s h i m s e lf , a s u c c e s s o r

w h o w o u l d f u l ly d e s e r v e t h e a c c o l a d e w h i c h W I N T N E R b e s t o w e d o n B O R EL . T h a t

w a s t o w a i t u n t i l 1 9 23 ; t h e s u c c e s s o r w a s t o b e H U G O S T E IN H A U S .

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