Some remarks on convergence in Orlicz space

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXI (1979) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE XXI (1979)

A. Kaminska and H. Hudzik (Poznan)

Some remarks on convergence in Orlicz space

1. Abstract. A Kozek in papers [4], [5] investigated Orlicz spaces L0 of functions defined on an abstract set T with values in a Banach space X , generated by an iV-function Ф:

X x T -+ [0, oo].

In this paper we shall prove that the d 2-condition is sufficient for equivalence of modular convergence and norm convergence in L0 . We prove also that if Ф{-, t) is continuous, ц is an atomless measure and X is separable, then the d 2-condition is necessary for equivalence of both above convergences. Moreover, we shall consider the relation between modular conver­

gence and convergence in measure ц.

We are indebted to Professor J. Musielak for his kind remarks in course of preparing of this paper.

2. Introduction. Let (T, Q, f i ) be a measure space, where T is an abstract set, g is a tr-algebra of subsets of the set T, ц is a tr-finite, positive, complete measure on Q. X is a Banach space.

2.1. Definition (see [4], [5]). A function Ф: X x T ^ > [0, oo] is called an N -function if it satisfies the following conditions:

There exists a set Txe Q, /2(7^7^) = 0, such that

(a) Ф(-, •) is M x Q - measurable, where $ denotes the er-algebra of Borel subsets of X;

(b) Ф (•, t) is lower semicontinuous on X for every t e Tx ; (c) Ф (•, t) is convex for every t e Tx ;

(d) Ф(0, t) = 0 and Ф (х ,0 = Ф( — x , t ) for every x e X , teT i;

(e) there exist ^-measurable functions a: (0, oo) and k: Tx -> (0, oo) such that \\x\\ ^ 2(r) implies Ф(х,г) ^ a (t) for every teT^;

(f) there exist /i-measurable functions ^q: Tx- > (0, oo) and 1: 7] - > (0, oo) such that \\x\\ ^ g(t) implies Ф{х, {) ^ ± (t) for every t e l i.

If Ф fulfils conditions (a)-(d), then Ф is called an N"-function; if it satisfies conditions (a)-(e), it is called an N ’-function.

R em ark. If Ф is an ^''-function and satisfies condition (e), then

— Prace Matematyczne 21.1

lim <2>(x,0= oo for every t e T x\ if it satisfies condition (f), then Ф(-, t) IWI-=o

is continuous at zero for every te 7 i .

Let M x be a set of all /i-measurable functions defined on T with values in X. The measurability of functions is understood as in [2 ].

By the Orlicz space L0 we mean the set of all functions х(-)е. Жх for which there exists a constant к > 0 such that

1ф(кх) = J <P(kx(t), t)dg < oo.

T

The functional I0 (x) defined on L0 is modular; L0 is a modular space (see [7], [8]).

The space L0 with Luxemburg norm is a Banach space. For other properties of Orlicz spaces L0 see [4], [5].

We denote by dom I0 the set I0 (x) < oo}.

2.2. Definition. We say that a sequence of functions x„(-)gL0is modular convergent to x ( - ) e L0 if there exists a constant к > 0 such that

lim I0 (k(xn — x)) = 0.

П -*OD

Norm convergence of a sequence of functions x„(-)eL0 to x{ - ) eL0 is equivalent to

lim I0 (l(xn — x)) = 0 for all l > 0.

П -*■ 00

The function Ф satisfies the A2-condition if

Ф(2х, t) < КФ( х , t) + h(t)

for all x e X and t e Tx, where К is some positive number and h (•) is an integrable non-negative function.

In Orlicz spaces L0 generated by N -function Ф: X -* R + without para­

meter t, the following d 2-conditions are applied (see [6]):

(d'2) 3 V Ф(2x) ^ КФ(х) if f i ( T ) = oo

K > 0 x e X

and for X = R l

(d2,vq) V V 3 Ф(Ъ>) s: КФ(с) if M(T) < 00.

К>0 up >0 ^{v q}

It is known that if g is atomless, then d 2-conditions are sufficient and necessary for equivalence of modular convergence and norm convergence in the case g T = oo and g T < oo, respectively. Obviously, the class of JV-func­

tions with parameter t contains the class of jV-functions without parameter.

Indeed, for any N -function Ф without parameter we can put Ф(х,г)

= р(г)Ф(х), where p(t) = 1.

If an N -function Ф without parameter satisfies (d2)-condition, then it satisfies also d 2-condition with h(t) = 0.

Convergence in Orlicz space 83

If p T < oo, X = R 1, and Ф satisfies (d2,^-condition, then for every v e R 1 we have <P(2v) ^ K<P(v)+ K<P(v0). Since p T < oo, jK<P(v0)d/j. < oo

7 so Ф satisfies d 2-condition with h(t) = КФ(и0).

Now, we note that there exists iV-function Ф{х,Ь) with parameter t which satisfies d 2-condition with h(t) ф 0 and it does not satisfy /^-condi­

tion with h(t) = 0. For example, let us put X = R 1 and let Ф(р,/)

= p(t) Ф1 (v), where Фх is N -function satisfying (/12 „^-condition, where vQ > 0 is the smalest number v such that Ф1{2и) ^ К Ф1(ю). Further, let p(t ) ф 0 and §p(t)dp < oo. Then for every u e [0 , oo) we have

T

Ф(2v, t) ^ КФ(с, г) + КФ(р0, t).

Since J КФ(и0, t)dp = КФ1 (и0) j p(t)dp < oo, the N-function Ф satisfies

7 T

d 2-condition with h(t) — КФ(и0, t). But Ф^т0) > 0 and p(t) ф 0, so /^-con­

dition cannot be fulfilled with h(t) = 0.

There exist N -functions Ф(х,г) with parameter t such that p T = oo and |Ф (х,г)^/7 < oo for every x e X and such that for p T < oo,

T

j Ф(х, t) dp = oo for x Ф 0. It suffices to take X = R 1, T = Rn, p -Lebesgue

t " _ 2 " 1

measure, Ф(х, t) = \ \ e tj and Ф(х, 0 —

П

j=i j=i N

the components of t = (fl 5 tn) and Ф1 is an N-function without parameter.

The condition J Ф(х, t) dp < oo for every fixed x e X and pA < oo is often

A

used. It means that simple functions belong to Orlicz spaces L0 ([7], see also condition A in [4]).

3. Results.

Proposition 3.1. Let us assume that p T < oo, Ф is an N"-function and Ф fulfils the condition

(e') V 3 IMI > ô => Ф(х, t) > a(t),

<5 > 0 a(-)

where a(-) is a measurable function such that a(f)e(0, oo) for every t e T r . Then 1Ф (x„) -> 0 implies x„ -*■ 0 in measure p.

P roof. Let x „ ()e # x and / ф(х „ )-» 0 . Assume, for a contrary, that *„(•) does not converge in measure to zero. Then there exist e0 > 0 and rj0 > 0 such that for every N there exists nN > N such that p { t e T: \\x„N(t)\\

> £o} > I/o- Let

An = {t e T : \\x„Nm > e0}.

Hence pAN > rj0 for every N. We are finding a set B œ. T and a constant К > 0 such that p(T\B ) < and Ф(х,г) > К for ||x|| > e0 and t e B .

In fact, let Bn = {t e T : a (t) > 1/n}, where a( ) is the function from condi- 00

tion (e') with 3 = e0. We have Bn a Bn + 1, (J Bn = T for every t e T t and

n = 1

lim pBn = pT. Since p T < oo, so lim p{T \B n) = 0. Consequently, there exists

M -*■ 00 П—► 00

a set Bno satisfying the inequality p(T\B „Q) < i ^ 0-

Let В = B„0 and К = 1 /п0. If t e B , then a (t) > К and from condition (e') we obtain Ф(х, t) > ot(t) > К for ||x|| > e0. Thus, for every N, we have (As r \B ) > jriо for all N, and

1ф (*nN) = f ф {*nN (t) , t ) d n > f Ф (x„N(t), t) dn

T i4jynB

^ X/i {An n B ) ^ К ~ ~ > 0 f°r every N . The above contradiction completes the proof of the proposition.

Proposition 3.2. Let Ф be an N "-function continuous in X , finite a.e.

on T and let X be a separable Banach space. I f for every sequence {*„(•)} ®=i satisfying 1ф(х„ (•)) 0 we have *;„(•)-►() in measure p, then the function Ф satisfies the condition

(e") V V (||x||

0)

P ro o f. Let Ф does not satisfy condition (e")- We shall find a sequence (•)}*= l modularly convergent to zero, which does not converge in measure to zero.

Let L = U { t e T: Ф{х, t) = 0} and Lm — U { t e T : Ф(х, t) = 0 } ,

ХФ0 IWlèl/m

00

where n = 1 , 2 , . . . We have L = y Lm. Let be a dense subset of

m= 1 oo

the set Am = { x e X : ||x|| ^ 1/mj and let Bnm = [j { t e T : Ф(х”, t) ^ 1 /«}.

k=

1

oo

Now we will prove the following inclusion Lm a f) Bnm for every

1 ^ n= 1

m = 1 , 2 , . . .

Let then m be an arbitrary index and t e L m. Then there exists an element x 0 of the set Am such that <£(x0,t) = 0. Since {х^}^°=1 is dense in the set Am, so there exists a subsequence {х*.},*! of the sequence {хк}Г^{= 1} such that ||x0 — Xjt*.|j 0. Hence and from continuity of Ф we have Ф(х£, t) 0. Thus Ф(х**., t) < 1/n for some index k{. Hence t e B nm.

00

If we write Bm = f | Bnm, then the inclusion Lm <= Bm is satisfied.

n=1

The set L cannot be of measure zero, because condition (e") is not fulfilled. Hence the sets Lm (for n > N 0 and some N 0) are not of measure

Convergence in Orlicz space 85

zero either. The sets Bm are measurable, moreover, they include Lm, so цВт > 0 for n > N 0. We can find a measurable set. C such that C c: BmQ, where m0 > N 0 and 0 < fxC < oo, by the cr-finitenesg of measure ц.

Writing a = 1 /m0 we define the following sequence of the set-valued functions

This familly of multifunctions has the following properties:

1° N n{t) is a closed set in X for every n = 1 , 2 , . . . , for all t e T x because Ф is lower ’semicontinous,

2° N n(t) Ф 0 for all t e C , n = 1 , 2 ,... For, let t e C c= BmQ. From definition of the set BmQ it follows that for every index n there exists an element x™ne X such that Ф(х*” , t) ^ 1 /n and ||х™и|| ^ a. Hence x™neN„(t).

3° The graph of N„ (we denote it by Gr (N n)) is J x Q-measurable, because

Gr (Nn) = {(x, t ) e X x T: x e N„ { t )}

= {(x, t ) e X x T : ||x|| ^ а л Ф{х, t) ^ 1 /п}

= {(x, t ) s X x T: Ф{х, t) ^ 1/n) n { { x e l : ||x|| ^ a } x T ] e M x Q since Ф(-,-) is 3 x Q-measurable.

Now, in view of Theorem 5.1, 5.2 in [3], we obtain the existence of the measurable functions y„(-) such that yn{ t ) e Nn{t) for t e C and for all n = 1 , 2 ,... Let

The functions x„(-) are measurable and ||x„(t)l! ^ a for all fe C , n = 1 ,2 ,...

Moreover,

So we have a sequence of functions which modularly converges to zero, but does not converge to zero in measure. This ends the proof.

R em arks. Г Condition (e") is equivalent to the following condition:

2° If X = R 1, then conditions (e') and (e") are equivalent. (e')=>(e") is obvious. Let now (e") be satisfied and let Ô > 0 be given. If |x| > <5, then Ф(х,г) = Ф(|x |,t) ^ Ф(<5, г) for every teT j. Hence condition (e') is fulfilled for а(-) = Ф(<5,.).

3° If fiT < o o , Ф is an ЛГ-function, then norm convergence in the N„(t) = { x e l : |x|| ^ а л Ф (х Д ) ^ 1/n}.

Ф (х,г) = 0<=>x = 0 for every t s T x.

Orlicz space L0 implies convergence in measure. This fact follows from the proof of Theorem 2.4 in [4].

4° Norm convergence in Orlicz space L0 , where the measure p is infinite and Ф is an N- function, does not imply convergence in measure.

The following simple example was given by Châtelain in [1]. X = R 1, T = R+ = [0 , oo ), p-Lebesgue measure, Ф(х, t) = x 2/t. Obviously, Ф is an iV-function with parameter. Let

jyJTjn if t e [ n + \ , n + 2), I 0 otherwise.

Then x„(-)eL0 . Let К ■> 0 be any positive number. Then n + 2 jç2

1Ф(К хп) = J Ф( Кхп(^, t)dp = J --- dt = - — 0.

Г n+l n n

Hence ||x„(-)||l* ^-*■ 0. But p { t e T : \xn(t)\ ^> 1} = 1 for n ⁼ 1 ,2 ,...; therefore x„( ) does not converge in measure.

Th e o r e m 3.3. I. I f Ф is an N"-function, Ф is continuous at zero, Ф satis­

fies (q') and Ф fulfils condition A2, then norm convergence and modular convergence are equivalent on L0 .

II. If Ф is an N''-function continuous at zero, X is separable, p, is an atomless measure and if norm convergence and modular convergence on L0 are equivalent, then Ф fulfils condition A2.

P roof. I. By definition, norm convergence implies modular convergence.

1° First we assume that p T < oo. Let хп(-)еЬф and / ф(х„)->>0. To establish the theorem it is enough to prove that / ф(2х„)->0.

Let Tn — [ t e T : Ф(х„(г), t) ^ h(t)}; then

1Ф{2хп) = {Ф (2xn{ t ) , t ) d p + J Ф(2xn(t), t)dp

Tn T \T n

^ ( K + 1) J Ф(х„(0, t ) d p + f ф (2xn(t), t)dp = J ln + J 2n-

T J n Tn

We have J l n~* 0 by assumption. Since / ф(х„)->0, so, by Proposition 3.1, x„(-)-> 0 in measure. Let us take any subsequence x (•) c xn(.). By the Riesz theorem, we can find a subsequence x„k (•) of the sequence x„fc(•) of functions which converge to zero almost everywhere. From the continuity of Ф at zero we have Ф(2x„k (t),t)^>0 for a.e. t e T . Moreover, if teT„, then УФ(2х„(0, t) ^ ( K + l)h(t). Hence J 2n -> 0, by the Lebesgue dominated

n ki

convergence theorem. Therefore, J 2n -+ 0.

Convergence in Orlicz space 87

2° Let fiT = оо and let e > 0 be given. Then for A <= T 1ф(2хн) = j<P(2xH(t),t)dn = S<P(2xn{ t ) , t ) dn+ J <P(2x„{t),t)dfi

T A T~,A

{Ф(2х„(0, t)dn + K

J

f

A T \A T \A

We choose a set Л in such a manner that J h(t)dpi < e and fxA < oo;

1 \A

then J 3 < e. By assumption we have J 2n-+ 0. Since /1,4 < oo we apply part 1° of this proof and we get J ln < e for n > N. Therefore, / ф(2х;„)-> 0.

II. To prove the theorem it is enough to show that dom / ф1 <= dom I02, where Ф1(х,г) = Ф{х, г) and Ф2(х,г) = Ф(2x, t ) (see Theorem 1.7 in [4];

A. Kozek established in that paper that under our assumptions the inclusion dom / ф1 c dom I02 implies condition A2).

Let us suppose that norm convergence is equivalent to modular con­

vergence, but there exists х ( ) е . Ж х such that x ( ) e d o m /01 and x(-)^dom i02.

Г There exists a set A с T such that цА > 0 and Ф(2x{t), t) = go

for all t e A . Since /i is atomless, there exists a sequence of sets Ane Q such that An a A, /u(A„) > 0 and /o4„->0. Let x„( ) = x(-)z^„(-)- We have / ф (*„) 0 but l 02 (2x„) = go, a contradiction.

2° Ф (2х(0, t) < oo for a.e. t e T .

(a) ц Т < oo. Let Ak = { t e T : Ф(2x(f), t) < к}; then A 1 с A2 c ..., and 00

U Ak differs from T at most by a set of measure zero. Since /tT < oo, к = 1

we have lim /i(T \A k) = 0. Now, let *„(■) = х(-)Хги„(-); then / ф(х„)-> 0, but 1ф(2хп) =

J

T An

for n = 1 , 2 , . . . , because j Ф(2x(t), t ) df i = oo and j Ф(2x(t), t)dpi ^ пцА„ < oo,

a contradiction. T A

(b) fiT = oo. We can suppose that |Ф (2 x( t ) , t ) dn < oo for all A e Q ,

A

fiA < co. Let £„->0. Since 1ф(х) < со, so there exist sets An such that

M „ < GO and

j

т\лп

Therefore, / ф(х„)->0. But

/ Ф(2х„) =

J

T \

This contradiction completes the proof.

0 0 .

References

[1] J. C h â te la in , These, 1974-1975, Université de Perpignan, Université des Sciences et Techniques du Languedoc.

[2] N. D u n fo rd and J. T. S ch w arz, Linear operators, Part I, New York 1958.

[3] C. J. H im m e lb e rg , Measurable relations, Fund. Math. 87 (1975), p. 53-72.

[4] A. K ozek, Orlicz spaces of functions with values in Banach spaces, Comm. Math. 19 (1976), p. 259-288.

[5] — Convex integral functionals on Orlicz spaces, preprint, May 1976.

[6] M. A. K r a s n o s e l’skiT, Ja. B. R u tic k iî, Convex functions and Orlicz spaces, Gosud.

Izdat. Fiz.-Mat. Literat., Moskva 1958 (in Russian).

[7] J. M u sie la k and W. O rlic z , On modular spaces, Studia Math. 18 (1959), p. 49-65.

[8] H. N a k a n o , Generalized modular spaces, ibidem 31 (1968), p. 439-449.