Eduard Emel'yanov

INVITATION TO ERGODIC THEORY

PrefaceThe ergodic theory goes back to the end of 19th centuryand has roots in physics (in particular in astronomy andin gas mechanics). Nowdays it is one of basic techniques inprobability theory, measurable dynamics, and in many otherareas of mathematics.

The main aim of this course is to give an introduction tobasic concepts of ergodic theory such as recurrence, ergod-icity, mixing, weak mixing, etc. It requires some knowledgeof measure theory and Lebesgue integration. We include ashort presentation of these topics in Chapter 1. Althoughthe material of this chapter is standard and may be found inmany textbooks (see, for example, 17,4,9,10]) we includefew proofs of interesting elementary theorems there. Weconsider recurrence and ergodic transformations of measurespaces, and study selected classical examples i n Chapter 2.

Then, in Chapter 3, we prove the Birkhoff individual ergodictheorem and the von Neumann mean ergodic theorem. Inthe last chapter, Chapter 4,, we discuss Frobenius - Perronoperators, mixing, exactness, and weak mixing.

We emphasize the introductionary character of this one-semester graduate course and send the reader for delicate,difficult and interesting topics of ergodic theory to textbooks

[3], l5l , l7l, [11]

2

Contents

Elements of Measure Theory

1.1. o-Algebras1.2. Measures1.3. Lebesguelntegration

Elements of Measurable Dynamics

2.1. Examples of tansformations2.2. Recurrence2.3. Ergodic Tbansformations

Ergodic Theorems

3.1. The Birkhoff Ergodic Theorem3.2. The von Neumann Mean Ergodic Theorem3.3. Applications of Ergodic Theorems

Related Topics

4.1. Flobenius - Perron operators4.2. Mixing

Bibliography

Chapter 1

Elements of Measure Theory

In this chapter, we outline briefly several essential conceptsof the measure theory and the Lebesgue integration. Forexhaustive presentation of these topics, we refer to standardtextbooks [1, 4, 9, 10].

1.1 o-Algebras

The theory of o-algebras plays a significant role in the prob-ability theory and in the ergodic theory.

1.1.1 Algebras and o-algebras of sets

Definition 1.1.1 A collect'ion\ of subsets of a set X iscalled an algebra i,fa)X€I;b)A€I+X\AeI;c)A,,BeI+AUBeI.

CHAPTER 1. ELEMENTS OF MEASURE THEORY

It is an easy exercise to show that any algebra is closed underoperations -4 n B and,4A B - (A \ B) U (B \ A)

Definition 1.L.2 An algebra D 'is called a o-algebra,f UAn €D yi,elds for any countable fami,ly {An}n g I.

k

As in the previous exercise, it is easy to show that any o-algebra is closed under countable intersections. A bit moredifficult (but stil trivial) exercise is to show that an algebra,which is ciosed under countable unions of pairwise dis.jointsequences of sets, is a o-algebra.

A simple example of a o-algebra is the collection P (X) ofali subsets of a nonempty set X.As an example of algebra which is not a o-algebra, one mayconsider the collection A(T,R) of all intervals (o,,b), (o,bl,lo,b),, lo,b1, (-oo, b) , (o,+oo) in IR and their finite unions.

Exercise 1.1. 1 Show that the fami,Iy of all countableand all co-countable subsets of a non-empty set,is a o-algebra. What would be 'in the case i,f finite subsets ,in-

stead of countable subsets are considered.

Given a non-empty family F of subsets of a non-empty set X,the algebra A(F) generated by F is the intersection ofall algebras containing F (at least one such an algebraexists, the algebra P (X)). The definition of the o-algebraA"(F) generated by F is similar.

1.1. r-ALGEBRAS 5

Exercise 1 .L.2 Show that A(7, IR) - A(F), whereF - {(-oo, bl}a.o u {{b}}a.o.

The o-algebra A,(A(T,R)) is called the Borel o-algebraof subsets of lR. More generally, the Borel o-algebra ofa topological space (X, r) is the o-algebr a Ao(r) generatedby the collection r of all open subsets of X .

It is often a challenging problem to describe the o-algebraA"(F) generated by a family .F in a more constructive waythan just the intersection of all a-algebras containing F.

In the next two subsections we study several important re-sults about o-algebras and recommend the reader to com-plete all details of the proofs below, or to consult textbooksIike [1 , 4, 6, I7].

1.L.2 Dynkin systems (optional)

Definition 1.1.3 A collect'ion K of subsets of X ,is sa,idto be o zr-system ,f i,t 'is closed underintersect,ion of angtwo of its sets, and A e K.

Denote by r(F) the zr'-system generated by F gP6)

6 CHAPTER 1. ELEMENTS OF MEASURE THEORY

Definition 1.L.4 A collect'ion D g P(X) is called a,

Dynkin system i/1) X eD;2)AeD+X\AeD;3) {A*}t'.xgD k An)Ap:aforklp impl'ies

UAn e D.k

Denote by D (F) the Dynkin system generated byF EP6)

Theorem 1 If K be a n-system then A"(K) -D(K).

Proof: It is obvious that D(K) q A"@) is a Dynkinsystem. We show that A,(K) g D(K)Note that any Dynkin system, which is a a"-system, isa o-algebra. So, it is enough to show that D(K) isa n'-system.

Let A € D(K) and consider the following family:

DA: {B e D(K): AnB e D(K)}.

The collection D,q, is a Dynkin system. Indeed, the condition1) of the last definition is obvious. Let {Bp}p q Da andBn) B^ : o when k I *. Then {An B*)n eD(K) is apairwise disjoint family. Hence

Ute n ail - An U Bn e D(K)

1.1. r-ALGEBRAS

and thenBn e Dn.

Then the condition (3) is satisfied.

To check the conditio" (2), take B e Da. By

An(x\ B):x\((An B)u (x\ A)),

Exercise 1.1.3 Checle the last formula carefully.

We obtainAn(x\B)eDa

since A n B € D(K), X \ A € D(K), and since(AnB) u(x\ A):sThus Dars a Dynkin system for any A e D(K)

Exercise 1.L.4 ShowthatAe K + K gDa.(ntnt Be K+Bn AeKcD(K)+aenn)

Let A € K. Using K g Dt, we obtainD(K) g Da

-c D(K), and hence D(K) - Da for every

A e K. Consider the foilowing implications:

AeK+DA:D(K)+li,f A e K and B e D(K) - DA, then A)B e D(K)I =+

lx q Dn e D@) i,f B e D(K)) +DB : D(K) (VB € D(K)) =+

A)B eD(K) (VA,B e D(K)).The last condition means that D(K) is closed under theintersection, what is required. I

Uk

8 CHAPTER 1. ELEMENTS OF MEASURE THEORY

1.1.3 The z--.\-Theorem and the Monotone Class Theorem(optional )

Let us consider now two more results of the same naturewhich are known as the zr-)-Theorem and the MonotoneClass Theorem.

Definition 1.L.5 A collect'ion L C P(X) 'is calleda ,\-system or \-class fo)XeL;b)A,Be L, AgB+B\AeL;c){An}ngL, AnIA+AeL.Denote by )(F) the ,\-system generated by F gP(X)

Theorern 2 (zr-,\-Theorem)U F is a r-system then A"(F) -

^(.F).Proof: Obviously .\(.F) g A"(F)To prove )(f) ) Ao(f), it suffices to show that )(f) is ao-algebra. It is enough to show that )(.F) is closed undercountable intersections. (WhA? Show thislConsider

)'(.r) -{Ae )(r) :AnBe^(.F) VBe f).Then

^tV) is a )-system (Why? Show this!) containing

F. Hence )t("r) - )(.r).Consider now

^rV)-{A€.\(r) :AnBe^(F) VBe )(-r)}

1.1. r-ALGEBRAS

Then ^r(F)

is (again as above) a ,\-system containi ng F(Hint: start w'ith x(F) instead of f and remark that)(^(4) - ^(r)).

Hence F e ^r(f)

g ,\(4 and there-fore )2 (F) - ^(F).

The last equality means that ,\(.F) isclosed under countable intersections, what is required). n

Definition 1.1.6 A fami,lA M g P(X) i,s called a mono-tone class i/i,)x#a;i,i,) {An}r g M, An I A + A e M;iii) {Br}r I M, Bn I B + B e M.

Denote by M(F) the monotone class generated byF CP6)

Exercise 1.1.5 Show that if a monotone class M ,is analgebra tlr,en M is a o-algebra.

Theorem 3 (Monotone Class Theorem)If A 'is an algebra then A"(A) : M(A).

Proof: Obviously M(A) S A"(A). To show the converseA"(A) g M(A), it is enough to prove that M(A) is analgebra.

For each G g X, define

MG:{BgX: B\G, BuG, G\B€M(/)}It is obvious that Ms is a monotone class (just becauseM(A)is)

10 CHAPTER 1. ELEMENTS OF MEASURE THEORY

Exercise 1.1.6 Show that E e A+ Ag Ms.

Thus M(A) e Mn for all E e A. Therefore

C e Mn (VE € A,C e M(A)),

which is equivalent to E e Mc (WhA? Show thi,sl.

So, for any C e M(A)), M(/)) q Ms (since A g Msand Ms is a monotone class). The condition

M(A) e Mc e M(A) (vc € M(/))means that for every B, C e M(A):

B\C, BuC, C\B €M(A).Together with the fact that X € M(A)., this implies thatM(A) is an algebra. The proof is complete. n

1.2. MEASURES

L.2 Measures

Definition 1.2.L Let A be a o-algebra of subsets of X.A funct'ion pr,: A- nR - R U{+*} ,is called, o measure,fo) p(a) - 0;

b) p(A))0forallA€A;c) t-r (? r-) : ? t @n) for any countabte fam,ity {An}nof pairwise disjo'int sets belong'ing to A.

In this course) we are interested mostly in "small measures" :

probabilistic (p(X) - 1), finite (p(X)_. oo),

or o-finite (={&}8, q "4 such that X - [J X" and

lr(X,) ( oo for each n). n':1

Definition 1.2.2 A tri,ple (X,A, p,) it called a measurespace.

Except of counting measures on countable sets, construct-ing a measure on a o-algebra could be a difficult task. Weshall not go deeply in details and only mention key steps ofconstruction of the Lebesgue measure on IR.

L.z.L Lebesgue measure on IR

First of all, we need a large enough a-algebra (containingthe Borel algebra 6(R)) otr which our measure will live.

11

12 CHAPTER 1. ELEMENI:S OF MEASURE THEORY

Secondly, this measure should agree with a certain functionon a certain subset of 6(R) (for example, with the length onthe collection J g 6(R) of all intervals).

For this purpose, w€ extend the length I . J -+ R+ -IR+ U {+*} to a function \ . P(R) * R1, and then takethe restriction of ) on a certain o-algebra D ) B(R), otrwhich ,\ acts as a measure. Some of details follows.

)(A) - inr {: ren): 11, e r k Asp.r}

I - {A, )(E) : \(E.A)+)(tr.A") for all E e 2(R)},where we denote A" - R \ A.

The classical Caratheodory theorem says that t is a o-algebra containing 6(R) and )(1) - l(I) for every I € J.Moreover, D is complete w.r.t. ) (i.e. )(B) - 0 =+ B e I)and ,\lr is a measure.

Remark that the Caratheodory theorem holds true if we startwith any countabiy additive nonnegative function on an al-gebra of sets (not necessary with the length of intervals).

Definition 1.2.3 The o-algebra D constructed abouecalled a Lebesgue algebra, and the n'Leasure )1"called o Lebesgue measure.

It is worth to note that the Caratheodory theorem is ratherdeep result and that the ideas that had led to the Lebesguemeasure space (R, I(lR),

^) go back to the end of 19th cen-

tury.

,is

,is

1.2. MEASURES

Remark that the Lebesgue algebra D(R) is strictly biggerthan the Borel algebra 6(lR). It can be shown (for example,by using of Monotone Class Theorem) that the cardinality ofthe Borel algebra Card(6(R)) is c - 2N. On the other hand,the ternary Cantor set C is Borel measurable, Card(C) : c,

and )(C) : 0. Therefore by the completeness of D w.r.t. ),all subsets of C belong to I and thus

Card(D) ) Cu'd(P(C)) :2') c - Card(B(R)).

Although Card(I) - Card(P(R)), the o-algebra I is aproper subalgebra of 2(R). By using the Axiom of Choice,it is rather easy to construct an example of a subset of IR.

which is not Lebesgue measurable.

L.2.2 Isomorphisms of measure spaces (optional)

Two measure spaces (X, A,,p) and (X' , A , p') are said to beisomorphic mod 0 if there exist measurable sets Xo e X ,

XI S X' of full measure (i.e., p,(X \Xr) - p'(X'\X6) -0) and a bijection Xo I ,'rsatisfying1) A € A' 1"6, <+ 0-'@) € Alxo,z) p(d-'(A)) - t''(A)YA e AlroWe call such a bijection O an isomorphism mod 0 or justisomorphism.

Example 1 .2.L Let F e [0, 1] be a closed set such that)(F) > 0. Defi,ne a n'Leasure p on Dlp by

p(A) - I(,4) l^@) (A e tlr).

13

14 CHAPTER 1. ELEMENTS OF MEASURE THEORY

Defineo,rnapQ:F---+ [0, 1] by

_ )(F n [0, r))XF)

d@)

It i,s not ltard to uerify that (f, f le,lt) 'is 'isomorpltic to(10, 1], Illo,tl, )) bg the 'isomorphi,sm Q.

More interesting fact is that any nonatomic completeseparable measure space is isomorphic to (1, Ilr, )), where1 g R is an interval (bounded or unbounded).

L.2.3 Lebesgue measure spaces (LMS)

Definition 1.2.4 A measure space (X,A, p) 'is sa'id tobe a canonical Lebesgue measure space, or s'implyLMS i/1) (X, A, p) 'is o -fini,te;2) X - D I I'is a d'isjoi,nt un'ion of a countable set Dand an'interual / q R such that (L,Alr, p) : (I ,t, )).The component (D, Alo,p) is called a canonical atomicLMS. The component (I ,Alr,F) (which is nothing else

than the interval -I equipped with the Lebesgue measure .\)is called a canonical nonatomic tMS.

Definition 1 .2.5 A complete measure space (X, A, p) iscalled a Lebesgue space if it 'is 'isomorph'ic mod 0 to acanon'ical LMS.

Remark that in the ergodic theory, Lebesgue spaces are ofmain interest among all measure spaces.

1.2. MEASURES

I.2.4 Approximation with semirings (optionar)

Definition 1.2.6 A nonernpty r-system R g P(X) iscalled a semiring on X if , for any A, B e R, the;e erist

disjoi,nt sets Et,...,En €R such that A \ B - lJ Er.k:l

As an example of semiring, one may consider the collectionJ(R) of all intervals in IR.

The following proposition is almost obvious.

Proposition L.z.L Let R be a sem'iring. If A - U An,n:I

where An e R, then A - i) ,r, where Cn e R.k:L

Proof: Define a sequence {8"}" by 81 - At, and for

n ) 1, Bn- An\(Aru. .. uA,-t). Then A - 1l f,. The

only thing that is left to write each B, asairi#":i union ofsets belonging to R. We show this by induction. Obviously,

m,p

81 € R. Assume that B, - l) D'u, where Den € R, for

p: I,,2,...1r1. Our aim is to Jfrl* that Bn+rsatisfies thesame property.

nnBy the construction of {Bn}n, U Ai : I) Bi. Hence

j:7 j:I

Bn+r:An+r\U Aj:An+r \U Bj:

15

j:I j:1

16 CHAPTER 1. ELEMENTS OF MEASURE THEORY

nnpj

-nAn+t\Bi-nl)o'-j--1' j:l k:1

for som "

Etk e R. We may also suppose that all p1 - / by

adding, if it is necessary, some E* - a. Thus

nnl'bn+t-nLlr'r-i:L k:7

lJ t E/,rn tr'*rn... n 4,, kt,kz,...,k, € {7,2,..., /}}which is nothing else than a disjoint union of elements of R.

By the induction, all Bn have such representation, whichcompletes the proof. n

Let us mention the following useful approximation propertywhich follows directly from Definition I.2.7 and Proposi-tion 7.2.7.

Proposition L.2.2 (First Littlewood Principle) Let(X, A, p) be a nxeasure space wi,th a suffic'ient sem'iring

Definition 1.2.7 Let (X,A, p) be a measure space. Asem'iring C g A of sets o/ finite measure is sa,id to be

a suffi.cient semiring for (X,A,p,) xf for euery A e A:

,r {i te): Ae|", , ci. c}t j:l i:7 )

1.2, MEASURES

C. Let A e A, p(A)

erists a fin'ite disjoi,nt un'ion

such tlt at

17

Cp of sets Cn € C

and let ep

Ao: lJk:I

p,(AAAo) < r.

Among others applications of the concept of the sufficientsemiring, w€ mention the following useful approximationlemma.

Lemma L.2.L Let (X,A, p) be a nleo,sure space w'ith asufficient sem'iring C. Then, for any A wi,th p(A) ( oo,there erists a sequence Hn I in A such that p(H") ( oo,H" I H ) A, p(H \ A) : 0, and each Hn 'is a countabledi,sjoi,nt union of elements of the sem'iring C.

Proof: Since C ts a sufficient semiring, for any s ) 0, there

exists H(r)- 3t,,Cj € C,such thatAq HG)andj:7

p(H(e) \ A) ( e. Write Hn: H(1) ) H(+) n . . . ) H(*)Since C is a n-system, it is easy to verify that each Hn ts acountable union of elements of C. By the Proposition I.2.L,Hn is a disjoint countable union of elements of C. Further-more, Hn I H, which obviously has the required proper-ties. n

n

18 CHAPTER 1. ELEMENTS OF MEASURE THEORY

1.3 Lebesgue Integration

In this section, we introduce thearbitrary measure space (X, A, p)properties.

1.3.1 Measurable functions

Lebesgue integral on an

and summarrze its main

(); e R).

Definition 1.3.1 A function f : X ---+ IR zs called mea-surable i,f f-'(I) e A for euery I € /(R).

Note that the collection M(X, IR) of all measurable real-valued functions on X is ciosed under all point-wise algebraicand lattice operations (like addition, multiplication, takingthe positive part, etc.). Moreover, M(X, R) is closed underthe point-wise convergence.

L.3.2 Construction of Lebesgue integral

In defining the integration on (X, A, p), ,t is natural to as-

sume that the integral of lla is equal to p,(A), and that theintegral preserves linear operations. This gives us the firststep in the constructing of the Lebesgue integral.

Definition 1.3.2 Let p,(X) < oo and f e M(X,R) be asimple funct'ion:

n

f (") - t \tn,qu

i,:l

1.3. LEBESGUE INTEGRAT/OAI

Then th,e integral of f is defi,ned by

It is also reasonable to have an integration that allows pass-

ing to uniform limits of integrands on sets of finite measure.So the second step of our construction is following

Definition 1 .3.3 Let p(X)bounded, and {g"}" be a sequence of si,mple funct,ionsconuerging to f uni,formly. Then

ffI f @) p,(d,r) :- Iim I g,@) 1t(d,r).

J n---+6 J

From now on we ailow the Lebesgue integral to take infi-nite values. To avoid minor technical difficulties in the nexttwo steps of our construction of the Lebesgue integral, w€consider non-negative functions.

Definition 1.3.4 Let p(X) < oo and0 < / € M(X,R).Then

where

19

^nI

I f @) t'@r) '- ! \p(At).Iv :_1

X L-T

I tol u@,),-;,* | r.@) p,(d,r)

XX

r*(*):{ r(.) li W}'1")=t'

20 CHAPTER 1. ELEMENTS OF MEASURE THEARY

Now we remote the condition p(X) ( oo, by using an ideasimilar to that one in the last definition.

Definition 1.3.5 Let0 < f e M(X,R). Then

{x"l p@r):- sup

{l r@) p(d*),p(A). *} ,

where

I r wl rr@*) ,: I na,(*) f (r) p,(d,r).

AX

The last step is nothing else than the following technicalagreement:

Definition 1.3.6 Let0 < f e M(X, lR). Tlr,en

I tf"l 1t(d,r):- [ f @) p,(d,r) - [ r@) p(dr),J r' \ /' \ ./ J / \

i,f at least one of integrals 'in the ri,ght-hand si,de i,s fini,te.

The construction of the Lebesgue integration on (X, A, p)is complete.

Remark that the last technical agreement, which allows us

to deal with integrals which are not necessarily finite, cannotbe used for complex-valued functions (or, more generally, forfunctions with values in IR" for n > 2).

1.3. LEBESGUE INTEGRATIOAi 21

In our course) we shall often make of use the collectionLr(X, A, p) of all Lebesgue integrable real-valued func-tions on X (i..., functions / such that f f (") p(dr) exists

and finite). It is obvious that Lr(p) - Lr(X,, A, p) is a vec-

tor space. To simplify notations, we shall often write I f ap

instead ot I f @) p(d")X

Exercise 1.3.1 Show that th,e quoti,ent spa,ce

L,(p) - Lt(p)t {r

. L,(p) , I tftdr,: ,}

'is normed space w.r.t. ll ll r : Lt(p) - R defi,ned by

il t/t il, '- I ,,t du

X

The Lebesgue integral has some important properties thatwe shall often use. We state them without proofs.

1.3.3 Properties of the Lebesgue integral

Theorern 4 (Monotone Convergence Theorem)Let {f"}" g M(X,IR) sofzsfy 0 S f" I f a.e. Then

rfI f dp-lim I f"dp. n

J NJXX

CHAPTER 1. ELEMENTS OF MEASURE THEORY

We also refer to the monotone convergence theorem as toMCT. Among other things, it allows the integration of point-wise convergent series with non-negative terms:

Theorem 5 (Fatou's Lemma) Let {f")n e M+(X, R).Then

Remark that the inequality in the Fatou lemma may bestrict. For example, if (X, A, p) - ((0, 1], t, )) and fr:, . [(0,*1 then liminf f n - 0, but [- f" dp - 1 for all n.

The situation will be changed drarrro{ti.ulty if we suppose asequence to be bounded by a Lebesgue integrable function.

Theorem 6 (Dominated Convergence Theorem)Let {f"}" q M(X,IR), g e Lt(p), and lf"l a s a.e. foralln. If f"(r) -- f (") a.e. then f e Lr(p) and

fIim I lf"- fldp- o. n

n_-* r*

We also refer to the dominated convergence theorem as toDCT.

/1rt- inf f,) d"u <timinf I f,ouJ'n'-nJXX

1.3. LEBESGUE INTEGRATIO.^{ 23

Theorern 7 (Jensen's Inequality) Let g : IR ---+ IR be

a convex funct'ion, 'i.e.:

s(ar+(1 -c-)a)<as(r) +(t -*)g(a)for all r,U e R, 0 1a < 1. Then

,({ r,,) = {gff)d,,

for any f e Lt(p). n

L.3.4 Product of measure spaces (optional)

Let (Xr, At, p) and (X2,, Az, pz) be measure spaces. Forsimplicity, we suppose measures pr , Fz to be finite.

Denote by R the following set

R'.- {At x Az: At e Ar, Az e Ar)of all measurable rectangles in X1 x Xz. Note that Ris a z-system. Denote by O the algebra generated by R.

Define the function pr t Fz on R by

l-4 & Uz(A, * Az) '.- ltL(Ar) ' pz(Az)

and extend it to O. This is possible because each elementsof Q is a disjoint union of finitely many elements of R. Nowwe show that the function h & Fz is countably additive onO. For this purpose it is enough to show that h E Fz tscountably additive on measurable rectangles.

D0 co

,6- - l) o".n:I

Define a functian fn: At ---+ R by

f,(,): { p,(AD

li :ei+

Thus, by the MCT,oooooof

D u' s t'r(B") - t pr(A?)' pr(AD - t I fndt, :n:l n:l ":, i?

24 CHAPTER 1. ELEMENTS OF MEASURE THEORY

LetB"-ATxA|,Ai € At,AT € Azforn- 0, 1,...such that

It is obvious thatoo

Dr"@):t'r@8)n:7

IE-'") dt"- t"@?) p'(Ag): FtE t"(Bo)

Thus Fr I Fz ts countably additive on the algebra O. Anapplication of the Caratheodory theorem gives the measure,\ on the o-algebra o(Ar* Az) which extends ht 1t2. Themeasure .\ will be denoted by the same symbol W g pz.

Definition 1.3.7The n'Leasure space (X, x X2, o(A1 x Az), h g p,2) ,is

called the product of the n'Leasure spaces (Xr, At, p)and (Xr,, Az,, pz).

1.3. LEBESGUE INTEGRAT/OAI 25

Theorem 8 (The F\rbini Theorem)Let f (rr, rz) be a real-ualued Lebesgue ,integrable func-t'ion on X1 x Xz. Then the second and the thi,rd iterated'integrals below make sense and enjoy the formula:

I",*r,f (*',rz)dtttu t": I*,(1.,f (*,,rz)dttz) or, -

L,Ur,f ("'rz)dttr) ou'' n

Theorem 9 (The Toneli Theorem)Let f (*r, rz) be o, real-ualued non-negat,iue measurable

funct'ion on X1 x X2. Then the funct'ions

fz(rr)- t f@r,rz)dpz k ft(*r)- f f@,,,r2)dt-rtJx, Jxr-

are At- and Az-rneasurable, respectiuely, and

I*,,r,f (*'' rz)dpt' & t" : lr,(lr,f (*'' rz)dttz) o" -

L'U"f (*''rz)dtt) o"' n

Example 1.3.1 The non-negatiui,ty cond,ition'in tlr,e Tonel,itheorem 'is essent'ial. Indeed, consider the measure space

CHAPTER 1. ELEMEN"S OF MEASURE THEORY

(N,2(N), )) , where \ i,s the count'ing n'Leasure. Then forthe funct'ion f (r,A) - ?l)"-, . I[1e+1>,>a] 'it h,olds:

oo oo oo oo

I I r@,y)drdy -o+r: I I tn,y)d,yd,r11 11

Remark that i,t is ea,sy to modi,fy the the construct'ion tomake f (*, il cont'inuous on R?.

1.3.5 The Radon - Nikodym theorem and the Riesz rep-resentation theorem

In this subsection we formulate two very useful classical the-orems of the theory of the Lebesgue integration. The firsttheorem deals with a representation of measures throughfunctions and the second one with a representation of posi-tive linear functionals through measures.

Let (X, A, p) be a measure space. A measure u on A is saidto be p-continuous if for every A e A

p(A) - 0 => "(A) - 0.

Theorem 10 (The Radon - Nikodim Theorem)Let (X,A, p) be a o-fini,te n'Leasure space, Iet u be a p"-

cont'inuous n'Leasure. Then there erists a non-negatiue

functi,on f e L,,(lt) such that

,(A):lfdt, (vAeA). n

A

1.3. LEBESGUE INTEGRATIOAI

Let X be a locaily compact metric space. For simplicity,one may think that X g lR'. BV Co(X) we denote thelinear space of all continuous functions X -* IR vanishing oninfinity.

Definition 1.3.8 A mappins d: Co(X) * IR zs calleda) a linear functional i,f d@ft + 0 fz) : ad(f t) + gdjz)for all a,0 € R, ft, fz e Co(X);b) a positive functional i/

f (") > 0 (Vr e x) =+ d(/) > 0

for all f e Co(X).

Theorem 11 (The Riesz Representation Theorem)

For eaery pos'it'iue l'inear funct'ional Q : Cg(X) ---+ IR, thereis a unique fini,te measure LLO on the Borel algebra B(X)such th,at

0(h) - h(") pro@r) (Vh e Co(x)). n

Remark that the measwe p6 in the Riesz representation the-orem is obviously finite. Moreover, it is clear that every finitemeasure LL on B(X), gives a rise to a positive linear func*tional d, on Co(X) by

or@) : I h@) p(dr) (vh € e(x)).X

IX

28 CHAPTER 1. ELEMENTS OF MEASURE THEORY