Chapter III

THE MONOTONE CONVERGENCE THEOREM, FATO,U S LEMMA, THE

LEBESGUE CONVERGENCE THEOREM AND UNIQUENESS OF THE

EXTENSION OF THE FUZZY LINEAR FUNCTIONAL

3.0. INTRODUCTION

In the previous chapter we have seen how the

fuzzy linear functional -c is extended from s to5 1 which is

the analogous form of the extension of the non negative

linear functional I from L to Li. In the case of the

crisp theory the next step is to show that the non negative

linear functional I on Li is a Daniell integral on L1. This

is established by means of the analogue of the monotone

convergence theorem ; then further prove the analogues for

the integral I of Fatou ' s lemma and the Lebesgue convergence

theorem which are considered to be very useful in the

development of the theory. Here we are establishing that

the fuzzy analogues of the monotone convergence theorem,

Fatot's lemma and the Lebesgue convergence theorem are

true under suitable assumptions . Also similar to the unique-

ness of the extension of I to LI we show that the extension

of t to si is unique.

3.1. THE MONOTONE CONVERGENCE THEOREM, FATOU'S LEMMA

AND LEBESGUE CONVERGENCE THEOREM.

Notation 3.1.1. For as sequence ^(f } of fuzzy pn a

30

in sl inf (fn)a = fa when f = inf fn and a = sup an n

What follows is the analogue of monotone convergence

theorem.

Proposition 3.1.2. Let J( fn)a l be an increasing sequencen

of fuzzy points in sl such that inf an > o and let

fa = lim ( fn )a . Then fa E sl if and only if lim -c((f )n n are

In this case -c (fa) = lim r ((fn)a ).n

Proof:

Necessity . Let {(fn) a }be an increasing sequence: andn

let fa = lim (fn )a . Then fa >, (fn)an

fa) =^(fa) >,-c((fn)a ) if 1?iti

nc(

If fa C sl then

(fn)a )n ni.e., given kt there exists n such that n >, n

-C ((fn)a ) >, kt with t=1 for every n >, 11 o thenn

z(fa) =-r-(f a) = - which implies fa Therefore if

fa E sl then lim (fn)a < 00•n

Sufficiency. Let lim z((fn)a ) < Since {(fn)a is anrY

increasing sequence,

CO

E ((fn+l)a _(fn)a )n=1 n+1 n nEl(fn+l- fn)min(a an n+

00E (f 't - f

n=1 n ^ n a

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CU

nEl (fn+l-fn))inf ai, i E N

= (f-fl)inf ai, iE N

= ga say where a = inf ai.

By lemma 2.1.8,

Co(ga) ((fn+l)a (fn)a )

n =l n+l n

00

L (c(fn+l)a --t(fn)a )n=l n+l n

00

n l ^((fn)an))E (-C(fn+1)a ^n+1

since (f ) E s, for every nWIi a

i.e., z (ga

limt((fn)an ) - c (fl)a1)

+^ ((fl)a ) ^< lim'C((fn)a )1 n

Since ga+(fl)a1 = fa, (fa} (ga + (fl)cc

<'c(g(x) +-Iz (f1)aI

)

1< lime ((fn) a )n

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We have (fn)a "< fa so thatn

i.e., -cz ((f,), ) =-c((f) )

i.e.

n a

, limc((fn)a ) '< -^l(fa),n

^((fn)a ) ^< fa)

<-c(fa), for(`n)a

n

But lim -.((f ) ) <c(n an f

Since Z (f) '<

i.e.,

limc(( fn)a ), c(fa) =c(fa) = lim c(( fn)n

fa E sl and -C (fa) = limc(( f)n

Corollary 3.1.3 . The functional is a fuzzy Daniell

functional on the vector lattice sl.

For, c (fa ) >, 0 for every f E s, , -c is a pos.itivi;Ulinear functional on sl and satisfies the definition of

fuzzy Daniell functional.

The next proposition is the analogue of Fatous

lemma,

Proposition 3.1.4. Let ^f`n)cc

nbe a seque nce ^... iVC

fuzzy points in s 1, where sup an >,Then the fuzzy points

inf(fna and lim (fn )a are in "s"1, if lim-C(( fn)ct ) <n n r

In this case 7(lim (fna .<l.m T((fn)a ) .n

Proof: Define

Then 4 (gn) Pn}

(gn)u = (fl )a ' (f1) a n (f 4n l 2 1,;an

n

is a sequence of nonnegative fuzzy points

in sl which decrease to g = inf(fn)a , i.e., g = inf fn n

33

and P - Ssup a ,1

(-gn)1-finT (-g),_^.

lim z ((-gn)1-pn) <

Proposition 3.1.2•

P means (i_ rt) 'l ----) so that

Also .c (( -gn)1-P ) ^,< 0; i.e.,n

00 Therefore (-g)1_0 E s1 by

Now we wil i. show-that g,6-

we-have g1 _p E s1 so that -t (g )1--o

>, 1.0 and .< < 1,

g, E s

gP -,< g1_0.

Since(-g)l-,= -(g)1

< Since g g,

Therefore -c (g13) .< z (

Let ( hi) Y in f (fn)a -rn : hen j(h,^)Y j is a

sequence of non negative fuzzy Pointsin which increases

to lim (f ) 1n an as Therefore,

(h^ ) Y "< (fn )a for e ,< n ;n

((he )Y )L

((ht ) Y4 )

i.e.,

<'t((fn)a )n

I,<

for e v<, n;

inf Z ((f ) )e.<n n an

lim t ((he )Y^ )< in c ((fn)a

lim z ((h^ )Y ) ^< 1im -^ (( f))n an

E si by Proposition

< CO

3.1.2.

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Since c ((hL)YL) .< inf -o((fn)an) for every

z ( sup inf (f )a

) \< sup inf z ((f ) )t 1,<n n n t P,<n n an

t (Lim (fn)a ) '< li.m z.((fn) )n n

Analogous to Lebesgue convergence theorem we have

the following proposition.

Proposition 3.105. Let {(fn )a I be a sequence of fuzzyn

points in sl and let there be a fuzzy point ga in

such that for all n we have (IfJ)a < g,. Then if

fa = lim (fn)a , t (fa) = lim -C((fn)a ).n n

Proof: i(fn)a +gp} is a sequence of non negative fuzzyn

points in sl and by Proposition 3.1,x,-,

11lim ((fn)a + gp) = fa + gP is in 9,n

Also,

since -c((fn)a +gp) \< 2-c(g^)n

c(fa+gp ) = z (lim ((fn)a + 9P))n

1< lim c((fn)a + g13)n

limZ((fn)a ) +--c(g).

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Since -c is a linear functional on sly

z' (fa + g^) = z (fa) + "c(g^). Therefore,

M (fa) lim z ((fn) an

gP (fn)a is also a sequence of non negativen

fuzzy points. Therefore,

-c (grfa) m (gO-(fn)a ))n

^< lim t(gP-(fn)a )n

=z(go) - urn -r- ((fn)a )n

_ (g ) _-t ( fa) \< t ( g,) - lim c (( fn)a ).n

i.e., lim t((fn) ) -,< t (fn

Thus t (fa) = lim t ((fn)a )n

3.2. UNIQUENESS OF THE EXTENSION OF -C ON s TO sI

Let sue be the class of all fuzzy points fa of F

which are the limit of a decreasing sequence of fuzzy

points (fn) a ^ with sup an >, 2 in su such thatn

t (fn)a ) < °° and lim -C((fn)a ) > -°On n

Lemma 3.2.1. If fa is any fuzzy point with (fa) finite

then there is a g, E sue such that fa 4< g, and C(fa)= t(g)0

3b

Proof: If .(fn)a ^ is a decreasing sequence ofn

fuzzy points with sup an >, in su then 1 (-fn),_aY

is an increasing sequence of fuzzy points in su

with lim ((-fn)l-a ) in su. Then by lemma 2.1.7

C(lim((-fn)1_an)) = z (lim(-f n)1--an) =z(lim(-^fn)1--an

Since f(-fn)1_a j is an increasing sequence in su,n

consequently in sl by Proposition 3.1.2,

lim ((--fn)1-(x ) E i. Therefore lim (fn)a E sln n

as in the proof of Proposition 3.1.5. Thus suet sl.

Let fa be any fuzzy point in r with

Z (fa) < 00. Then for a given n, there exists (hn) E s^^Yr

such that

z (fa) = inf t((hn)Y )

aC<(hnYn

ioeo , t ((hn)1 ) < E (fa) + (n) , where (n) E Ti with t--1n t t

Define

( gnt3n

Y2(hl)Y1A (h2) ... (hn)Yn

_ (h '\h n o.. nh )1 2 n max (y1,Y2,..., Yn

37

Therefore fa \< (gn)O ,< (hn)1 and f (gnn n

decreasing sequence of fuzzy points in su with

Z(f < =((9n)O ) <t.((hn)Y ) ;n n

i.e., z (fa) t((9n)^ ) ^< z f +n t

s a

Therefore lim `C ((gn)On = z ( fa); i.e., lim'c (gn)u

Onexists and urn -C( (gn)O ) = t (lim(gn)O ) _ -c (g,)

n n

Since gp is the limit of a decreasing sequence ^ (gn

of fuzzy points in su, gP E suL Thus z (fa) . -r-(gP

Definition 3.2.2. A fuzzy point fa in F is said to be

fuzzy null point if fa E sl and Z ((IfI)a) = 0, where

a E (0,1] and 0 means fuzzy singleton 0.

Remark 3.2.3. If fa is a fuzzy null point and (Igl)

then 0,< t ((191)P) '< Z ((Ig))0) \<-c (fa) = 0;

i.e., z ((1 91)0) = z ((1 91)P) = -c ((IgI)p) = 0

i.e., g, E sl and g, is a fuzzy null point.

Proposition 3.2.4. A fuzzy point fa in F is in s1

if and only if fa = g, -h1, where a = min(0,y), gpE 5u

and hY is a non negative fuzzy null point . A fuzzy point

of F, hY is a fuzzy null point if and only if there is a

fuzzy null point k5 in suL such that ( Ihj)Y ^< ks

38

Proof:

Necessity. Let fa = go-h1 where a = min(3,y),

9P E su L and hY a non negative fuzzy null point. Since

g^ E su t and su 2 c sl, ga E sl. Also since hY is a nor:

negative fuzzy null point, hY E s Now,

(g+(-h))min (p,y) = fa F l.

Let k5 be a fuzzy null point in suk and

(Ihl) Y ^< k5 . Then z((Ihl ) Y ) =t(ks) = 0. From the

above remark hYE 1 and hY is a fuzzy null point.

Sufficiency. Let fa E sl. Then-c (fa) = c(fa) is finite

and there exists a gP E Su f- with fa < g, and z(fa) = t.(gp) .

Therefore , by = gp -fa is a non negative fuzzy point.

Also , t (h1) = t (gs-fa) =-c ( gp) - T-(fa ) = 0 Therefore,

hY is a fuzzy null point.

Let hY be a fuzzy null point . Then h-rEs1,

and -C ((I hl)y) = 0. Therefore by lemma 3.2.1, there is

a fuzzy point ks E sup such that

((^hI )Y) = c ( ks) = 0^

(!h,)Y < kj and

The following proposition establishes the uniqueness

of the extension of -c on s, .

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Proposition 3.2.5. Let-r- be a fuzzy Daniell integral

on a fuzzy vector lattice s and ^ be a fuzzy Daniell

integral on a fuzzy vector lattice t Ds.

If -c (fa ) = ^ (fa) I fa E -s then -ti D sl and C (fa)_^ (fa

for every fa E sl .

Proof; Suppose that { (fn)a be an increasing sequenc e

of fuzzy points in sl and let fa = lirn (fn)an. By

Proposition 3.1.2, fa E sl and t(fa) = lim t ((fn)a )n

lim ^ ((fn) a (fa) . We have shown that the fuzzyn

Daniell integral t *,a can be extended to sl by

Proposition 2.2.3 and in the same way on t can be

extended to t1. From above fa E sl implies fa E tl.

Hence s1 c tl o

Using proposition 3.1.2. sue. c. sl and from above

s1 c tl we get sue, C sl c tl. Therefore if faC sl then

fa E tl and z (fa) _ ^ (fa) for every faC sl.