Orlicz spaces of vector functions generated by a family of measures

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXII (1981) ROCZN1K! POLSK1EGO TOWARZYSTWA MATEMATYCZNEGO

Séria 1: PRACE MATEMATYCZNE XXII (19811

L.

and A.

(Poznan)

Orlicz spaces of vector functions generated by a family of measures

In the present paper the authors introduce a modular /ф defined on the space of measurable vector valued functions. The modular space of the Orlicz type ЬФ considered here is a generalization of the spaces investigated by A. Kozek in [7], [ 8] and J. Musielak and A. Waszak in [11]. Many results contained in [ 6] are improved in this paper.

1. Preliminaries. The notation and terminology introduced below will be used in the sequel without any further reference.

Let Г be a set, Q a cr-algebra of subsets of T, and M a family of countably additive non-negative measures on Q. Then the set function

VM

: <2 — CO , oo] defined by

vM{A) = sup fi{A)

IueM

is a (7-subadditive submeasure

[2],

vM

vM( 0)

0

( U Ai) ^ X

(A

) f°r all

(A„)

Q. We

Qc

n n

ring of all sets A in Q on which vM is order continuous, i.e. vM (A„) -*■ 0 for every sequence (An) of subsets of A belonging to Q and such that A ,> 0 that is, A x

A 2

... and f] A„ = 0. Sets A e Q with vM(A) = 0 are called

ft

null. A property is said to take place almost everywhere (a.e.) in T, or for almost all (a.a.) t e T , if it occurs for all teT\A, where A is a null set.

Let X = ( X , || • II) be a real Banach space. A function x: T-+ X is said to be measurable if

(i) x is essentially separably valued, i.e., there exists a null set A such that x(T\A) is a separable subset of X, and

(ii) x ~1 (G) 6 Q for every open (or Borel) subset G of X .

In particular, if x assumes only a finite number of values (on sets belonging

to Q), it is called a simple function.

176 L. D r e w n o w s k i and A. Ka mi r i s k a

We denote by S(X ) = S ( T , Q , M ; X) the linear space obtained from the set of all measurable functions x : Г-> X by identifying functions which are equal a.e.

Remark. Consider a family of complete measure spaces (T, Q^, p), set Q — P) Qfi> and define M to be the family of restrictions of the measures ft to Q. Suppose that M consists of er-finite measures and that either X is separable or M is countable. Then a function x: T ^ X is measurable (in the above sense) iff it is ^-measurable [3] for all p.

If x ,x „e S p O , n = 1,2,..., then x„ -> x (vM) means that the sequence (x„) converges to x in submeasure vM, i.e.,

vM{£: ||x(£)-x„(t)ll ^ e) -► 0 as n -> oo for every e > 0.

We shall use some relations between the convergence a.e., in submeasure ,vM, and uniform, collected in the proposition below. We omit the proofs, since they are rather easy modifications of the standard proofs (e.g., those in [3]); a detailed discussion was carried out in [1], Chapter III. We note only that, as a rule, the sets in Qc play the role of sets of finite measure.

1.1.

(1) Suppose T is a countable union of sets from Qc.

Then a function x : T X is measurable iff there exists a sequence of simple functions converging to x a.e. in T.

(2) Let a sequence (x„) cz S(X) satisfy the Cauchy condition in submeasure vM, i.e., xn — xm —>0 (vM) as m, n -> oo. Then there are x e S ( X ) and a sub­

sequence (y„) of (x„) such that

x„ -*■ x (vM) and y„ (t) -> x (t) for a.a. t e T .

(3) (The Egoroff Theorem.) Let x, xne S( X ) , n = l , 2 , . . . , x „ - + x a.e. in T, and let A e Q c. Then for every Ô > 0 there exists a subset В of A such that vM{B) ^ Ô and xn( t ) - * x ( t ) uniformly for teA\B.

All of the above extends, with suitable modifications and restrictions, to functions/: T -> [0 ,o o ]; we call such functions non-negative.

If / is a non-negative measurable function on T, we define f f dM = f f ( t ) d M ( t ) = sup f fdp, A e Q .

А А ц е М A

We shall usually write j f dM instead of j f dM; clearly j fdM = J fxAdM,

T A

where X

is the characteristic function of A. We note the basic properties

of the “ integral” j fdM:

Orlicz spaces o f vector functions 177

1.2. P

. Let f ,g ,f „ (n = 1,2,...) be non-negative measurable functions on T. Then:

(1) J fdM = 0 o / = 0 a.e.

(2) / ^ g a.e. => J fdM ^ J gdM; J (/ + 0)dM ^ $fdM + J gdM . (3) J (lim inf/„)dM ^ lim inf $f„dM; f n/ f a.e. => \fnd M /\fdM.

« - > o o « - > oo

00 oo

(4) И I / . ) « « X f / . « ;

n =1 n= 1

(5) J/„ dM -► 0 =>/„ -► 0 (vM);

(6) The set function Л и | fdM is a submeasure on Q satisfying An/A

A

=> J fdM / j fdM (hence it is o-subadditive).

A„ A

A submeasure v: Q -> [0, oo] is said to be vM-continuous with respect to a subring sé c: Q, shortly: (s i ,v M)-continuous, if for every

> 0 there exist A e s i and Ô > 0 such that v(T\A) ^ e and v(B) ^

if В a A and V

(B) ^ Ô.

1.3. P

. Let (/„) be a sequence of non-negative measurable functions on T, converging to 0 a.e. in T. Suppose also that there exists

a measurable function g such that:

(i) f n < g a.e. for all n,

(ii) J min (1 ,g )d M < со, for every A e Q c,

A

(iii) the submeasure v(A) = J gdM (A e Q ) is (Qc, vM)~continuous.

A

Then lim j f„dM = 0.

« - > o o

P ro of. Given e > 0, choose A e Q c and Ô > 0 as in the definition of ( s i , vM)-continuity for s i = Qc. By the Egoroff theorem, we find a subset В of A so that vM (B) < ô and f„(t)/u(t)^> 0 uniformly for teA\B, where и = m in (l,g ) and 0/0 is understood as 0. Let n0 be such that

/„(f) ^

(f)(l Ч- j udM)~1 for te A\B, n ^ n0.

A

Then

i f . d M < J f ndM + i f ni M + \ f ndM

T T\ A в A\B

^

(T\A) +

(B) +

J udM )(l + lu d M y 1

A\B A

^ 3e for n > n0,

which concludes the proof.

178 L. D r e w n o w s k i and A. K a m i n s k a

A function Ф : X x Г-* [0, oo] is called an . I '-function if (a) Ф is S x (7-measurable {S is the Borel ст-algebra in X), and

(b) for a.a. te T the function Ф(-, t): X -> [0, oo] is lower semicontinuous, convex, even: Ф( — x ,t) = Ф(х,Ь) for every x e X , and Ф(0,г) = 0.

1.4. P

. Consider the following additional assumptions on an ( '-function Ф:

(c) lim Ф(х, t) = oo for a.a te T .

(c') There exists a measurable function a: T -> (0, oo) such that, for a.a.

te T , if x e X and ||x|| ^ a(t), then Ф (х ,0 ^ 1.

(c") For every measurable function f : T -> (0, oo) there exists a measurable function cr. T - * (0, oo) so that, for a.a. te T , if x e X and ||x|| ^ a(t), then ф ( * , 0 ^ №

(d) For a.a. t e T and all r > 0,

inf Ф (x , t) > 0.

ll*ll = r

(d') For a.a. t e T and all x e X ,

Ф(х, t) = 0 <=>x = 0.

Then (c " )o (c ') => (c) <= (d) => (d'); moreover, (c)=>(c') if X is separable, and (d') =?> (d) if dim X < oo.

Proof. (c')=^(c"): If a is as in (c'), then a = max (a, af) is as required in (c").

(c) (c ) if X is separable: The following argument has been kindly communicated to us by A. Kozek. We may and will assume that the exceptional null sets in (a)-(c) are empty. The multifunction 11-> At

= [ x e l : Ф(х,Г) ^ 1} has non-empty closed values (by the lower semi­

continuity of Ф (-,£)), and its graph in X x T is .M xQ -measurable. Hence there exists a sequence (x„) of measurable functions such that (x„(t): ne IS}

is a dense subset of A, for all t e T (see [5]). Then the function a(t)

= sup ||x„(t)||+Я (X any positive number) satisfies (c').

n

(d') => (d) if dim X < oo : For a.a. t e T and any r > 0, the lower semi­

continuous function хь-»Ф (х,г) assumes its infimum on the compact set • (x e X : ||x|| = r ], and this infimum must be positive by (d')-

The remaining implications are trivial.

An A '-function Ф satisfying condition (c) in 1.4 is called an . 1 -function.

(We slightly depart from the terminology used in [7] and [8].)

Remark. It will be seen in what follows that, actually, it would suffice for our purposes to require that the restrictions of Ф to the products Y xT,

Y a separable subspace of X , have suitable properties.

Orlicz spaces o f vector functions 179

2. The Orlicz space L 0 (T, Q, M ; X ). Let Ф be an ,/F'-function. Then the functional

defined by

/ = I 0 : S ( X ) ^ [ 0 , oo]

I(x ) = J <*>(x(r),t)rfM(t)

is a convex pseudomodular [10] on S(2Q. From the lower semicontinuity of the functions Ф(-, t) and Proposition 1.2(3) it follows easily that I is lower semicontinuous with respect to the convergence a.e., that is:

(lsc) I f x, x „eS {X ) and x „-+ x a.e., then I (x ) ^ liminf/(x„).

П-* 00

In particular, for every x e S (X) the function r h> /(rx) is non-decreasing and left-continuous for r > 0. Using the modular I we define a (Luxemburg) seminorm N - N 0 on S(2f), possibly admitting the value +oo, by

N (x) = inf {r > 0: I(x/r) ^ 1}.

The pseudomodular I (or the seminorm N ) determines the Orlicz type linear space

L 0 = L 0 (T ,Q ,M ; X ) = (x e S (X ): lim/(rx) = 0}

r - »0

= (

S(X): I(r x ) < oo for some r > 0 }

= {x e S (X ): N (x ) < oo).

In what follows L 0 is always considered to be endowed with the seminorm N.

Note that if (x„) <= L 0, then iV(x„)-> 0 if and only if I(r x n)-> 0 as n-> oo, for all r > 0.

Remark. For each ц е М consider the corresponding pseudomodular /Дх) = J Ф (x (t), t)dfi(t), Luxemburg seminorm iV^(x) = inf (r > 0: /д(х/г)

r

^ 1} (xeS(2T)!), and Orlicz space Ьф{ц) = {x e S (X ): N^(x) < oo}. Since /(x) = sup 1ц{х) for all x e S (X ), we have also N (x) = supiV^^x) for all

x g

S(X ), whence

L * c f) L 0 {n) and L0 = (xeS(2Q: supiV^(x) <

}.

це M цеМ

This observation links our construction with that of Rosenberg [12].

From now on Ф is assumed to be an .1 -function; then N is a norm, as shown in Theorem 2.2 below.

For simplicity, but without loss of generality, we shall assume in proofs that the exceptional null sets appearing in various conditions imposed on Ф are empty.

3 — Roczniki PTM — Prace Matematyczne XXII

180 L. D r e w n o w s k i and A. Ka mi r i s k a

2.1.

Suppose X is separable and let a: Г-> (0, oo) be a measurable function as in condition (c') of Proposition 1.4 (existing by the implicaticn (c)=>(c') of Proposition (1.4). I f (xn) <= L 0 and iV(x„)->0, then (x„/a) -> 0(vM).

P roof. Let r > 0 be given and define An = (fe T : ||xn(t)|| ^ ra(t)}, neN.

If t e A n, then ||r~1 x„(t)|| ^ a(t), and so Ф (г~ г x„(t), t) ^ 1, by (c'). Hence vM(4„) < J Ф (г~ 1 x „ (t),t)d M it) ^ /(r-1 x „ )- »0 as «-►oo.

2.2.

77ie Or/icz spncc = L 0 (T, Q, M ; X) is a Banach space in which the norm convergence implies the convergence in submeasure vM on sets from Qc.

P roof. We first show that the seminorm N is a norm. Suppose N (x) = 0;

then /(/cx) = 0 for every k > 0. Hence, by Proposition 1.2(1), the union A of the sets (t: Ф(Ьх(1),1) > 0 } , ZeeiV, is a null set. On the other hand, since Ф is an Ж -function, Ф(/сх(1), t) -> oo as k -> oo if x (t) ф 0, and so A — { t : x (t) Ф 0}. Thus x = 0 a.e.

In the rest of the proof we shall deal with sequences of functions in L 0, only. Their joint range is essentially separable by our definition of measurability, and therefore we may assume X to be separable. Let a function a( ) be as in Lemma 2.1.

Suppose (x„) is a Cauchy sequence in L 0 . From Lemma 2.1 we have (x„ — xm)/a -> 0(vM) as m ,n->oo. By Proposition 1.1(2) there exist x e S (X ) and a subsequence (y„) of (x„) such that yn -+ x a.e. Then, by the lower semicontinuity of I stated in (lsc), we obtain

I (r(x — x„)) ^ lim inf / (r (ym — x„)) -►0 as n -> oo,

oo

for every r > 0. Thus x e L 0 and N (x — x„) -► 0 as n -► oo. This proves that L 0 is complete.

Now suppose (x„) с 1 ф, JV(x„)->0, and let A e Q c. Let us choose arbitrarily e > 0 and ц > 0. Since vM is order continuous on A and Am = {te A : a(t) ^ m }N 0, there is m such that vM(Am) ^ rj. Then we have

vM {te A : ||x„(t)ll ^ e} ^ vM {te A : Цх„(t)ll ^ b « ( 0 > m} + + vM { t e A : ||x„(0ll ^ e,a (t) < m}

< _{V + vM{t:} Их.(Oil ^ (e/m)a(t)}.

Since the last term tends to zero as n -* oo (by Lemma 2.1), it follows that vM {te A : ||xB(0ll ^ e) ^ 2rj for all sufficiently large n.

2.3. P

. Let Ф satisfy condition (d) of Proposition 1.4 and let (x„) c; L 0 and /(x„)-> 0. Then

(1) x„-> 0 (vM) on sets from Qc, and

(2) a subsequence (y„) of (xn) converges to zero a.e.

Orlicz spaces o f vector functions 181

P roof. We first prove (2). Choose a subsequence (v„) of (x„) so that oo

£ I(y „) < oo. Since

n= 1.

00 00

И Z

7' n = 1 n = 1

by Proposition 1.2(4), it follows that 00

Z ф(упи), t) <

for a.a. teT.

n= 1

Hence Ф(у„(г), t) -» 0 for a.a. teT , and now (d) implies -> 0 a.e.

In order to prove (1) we use (2) to see that every subsequence (x^,) of (x„) has a further subsequence (x") with x "-> 0 a.e.; a fortiori x" ->0(vM) on sets A e Q c, by Proposition 1.1(3). Hence x„->0 (vM) on sets from Qc.

The above proposition has a partial converse (which is also valid for Ф an J '-function).

2.4. P

. Suppose X is separable, Ф(-,г) is continuous for a.a.

te T , and whenever vM(A) > 0, then 0 < v M(B) < oo for a measurable subset В of A. Suppose also that every sequence (x„) in L 0 with I(x„) -> 0 has a subsequence converging to zero a.e. Then Ф satisfies condition (d) of Proposition 1.4.

P ro of. Fix an r > 0 and let {dl ,d 2,--.} be a countable dense subset of the sphere ( x e l : ||x|| = r}. Then

f (t) := inf Ф(х, t) — inf Ф ^ п, t), te T ,

||x|| = r n

is a measurable function on T (the second equality is justified by continuity of Ф(-, f)). Assume that vM(A) > 0, where A = {t: f (t) = 0}. Then, by our assumption on vM, there is a measurable subset В of A with 0 < vM(B) < oo.

Choose p0e M so that 0 < g0 (B) < oo.

It is easy to define for each ne IS a measurable simple function x„: T —> {dx,..., d„} ci X

so that

Ф(х„(0, t) = inf Ф (di, t), teT .

i ^ и

Clearly, Ф(х„(г), t) N / (t) as n -» oo for all te T . Since 0 < p0(B) < oo, by the Egoroff theorem there exists a subset C of В with p0(C) > 0 such that Ф (x„(t), t)-> 0 uniformly for te C . This and 0 < vM(C) < oo imply that I (x nXc)-+ 0- However, since ||x„(t)|| = r > 0 for all n eN and te T , no subsequence of (x„Xc) converges to zero a.e., contrary to our assumption.

Thus f (t) > 0 for a.a. te T , i.e. condition (d) is satisfied.

182 L. D r e w n o w s k i and A. K a m i n s k a

We say that Ф satisfies the A2-condition (cf. [8]) if there exist a constant к > 0 and a non-negative measurable function h on T such that

J hdM < oo and Ф{2x кФ(х, t) + h(t) for all x e X and a.a. t e T .

T

2.5.

Let Ф be an ,^-function such that (i) Ф(-,1) is continuous at 0 for a.a. te T ,

(ii) Ф satisfies condition (d) of Proposition 1.4,

(iii) Ф satisfies the A2-condition with a function h such that the submeasure

\{A) = J hdM, A e Q , is (Qc, vM)-continuous.

A

Then the norm and modular convergence in L 0 are equivalent, i.e., if (x„) c= L 0, then N (x„) -* O o I (x „ )- > 0.

P ro of. We have only to show that /(x„)->-0 implies N (x n)- * 0 . Indeed, as easily seen, it suffices to prove that if /(x„)-*0 , then I(2y„)->0 for a subsequence (y„) of (x„).

Suppose I ( x n)-> 0; by Proposition 2.3(2) a subsequence (y„) of (x„) converges to zero a.e. Setting An = [t: Ф(уп(t),t) ^ h(t)} and using the A

-condition we get

(») I(2yJ sS (к + 1)Цуп) + J <P{2yn(t), t )XAn{ t)iM (t).

T

In view of (i) we have

^ (2yn (t), t) -> 0 as n -> 00, for a.a. te T , and the A2 -condition implies

Ф(2у„(0, t)xAn(t) ^ (k + l)h (t) for all ne IS and a.a. te T .

It follows, by Proposition 1.3, that the second term on the right-hand side of (*) tends to zero as n->oo, whence /(2y„)->0 as oo.

3. The subspace E0 {sé\ Let sé be a subring of the ring Qc such that if В c Ae.sé and B eQ , then Besé. We say that х е Ь ф has {sé, vM)-conti­

nuous norm if the submeasure Ah+ N (x%A), A e Q , is {sé, v^-continuous;

equivalently, if the submeasure d h / ( r x ^ ) is {sé,, vM)-continuous for every r > 0. It is clear that

A0 {sé) := ( х е Т ф: x has {sé, vM)-continuous norm}

is a closed linear subspace of L 0. Note also that the subspace { x e A 0 {sé):

supp x e s é j is dense in A0 (sé).

Assume that sé is such that

( + ) x%A e A 0 {,<é) fo r a ll x e X an d A e s é ,

Orlicz spaces o f vector functions 183

and let

E0 {sé) = Lin {

- x e X , A estf}t0 ;

thus Еф(,sé) is the closure in L 0 of the subspace of ^-simple functions.

Evidently, E0 {srf) c= A0 (jrf). Moreover, if x e £ 0 (j/), then I(r x ) < oo for all r > 0. Our next result gives a sufficient condition for the equality E o W

= АФ('Я/) to hold.

3.1.

Suppose

satisfies conditions

and

( + + ) for every A esé there is r > 0 and a non-negative measurable function f A on A such that \ fAdM < oo and Ф(х, t X f A(t) far ||x|| r

A

and a.a. te A .

Then E0 (stf) — A 0 (jtf).

P roof. Let x e A 0 (s/) and take any e > 0. Then there is A esé and Ô > 0 such that N (

\

) ^ £ and N (

) ^ e if В a A and vM(B) ^ 3.

Since A e Q c, by Proposition 1.1(1) there is a sequence (s„) of simple measurable functions vanishing outside A and such that s„ (f) -* x (t) for a.a. te A .

Since sé is hereditary with respect to Q and satisfies ( + ), we have s„eE0 (sé), for all n e N . By the Egoroff Theorem 1.1(3), there is a measurable subset В of A such that vM(A\B) ^ 3 and

d„ = sup ||

(t) — sn(t)\\ -*■ 0 as n ->■ со.

teB

To this В (Besé) choose r > 0 and f B according to ( + + ) (for В in place of A).

Let к > 0. Then

fc(x(0 -s„(0 ) = r l kdn[ r ( x ( t ) - s n(t))dn *]

and hence, using convexity of Ф and ( + + ) , we see that if r~ 1kdn ^ 1, then

ф(/с(х(г) — sn(t)), t) < r-1 kd„fB(t) for a.a. teB . It follows that

I(k (x -s „)x B )-+ Q as n -*■ oo for every к > 0 and so N ((x — s„)X

) 0 as n -> oo.

Let n be such that N ((* — sn) X

) ^ £- Then

N (x- s„ Xb) < N {x x t\a) + N (x xA\b) + N ( (x- sh)x b] ^ 3e

and s„xBeE 0 (sé).

It follows that x e E 0 {sé).

3.2. Theorem.

Suppose sé is non-trivial (vM(C)

for some Ce,<é)

and satisfies conditions ( + ) and ( + + ) , and that X Ф {0}.

184 L. D r e w n o w s k i and A. K a m i n s k a

Then Еф (s i) = Аф (s i) is separable if and only if the following conditions are fulfilled:

(i) X is separable.

(ii) s i is separable under the pseudometric

(A, B) = vM(A — B), A - B : = (A \B )u(B \A ).

(iii) There is a sequence ( B„) in s i such that if B0 = (J Bn, then П

A e s i =>ум (Л\В0) = 0.

P roof. Suppose E0 (s i) is separable and fix C e s i with vM(C) > 0 and x0e X with ||x0|| = 1.

(i) follows from the fact that X is the image of the (separable) subset { xxc•' x e X } of E0 (s i) under the map x x c ^ -x , which is continuous by Theorem 2.2.

(iii) Choose a sequence (Bn) in s i so that {хо/Св„: n e N } is dense in

( x 0

X

• A e s i], denote B0 =

Bn, and let A e s i. Then П

N (x0 X

\

0) ^ N (*o X

~ x 0 X

„) for all n e N - which implies

= 0 a.e., and so vM{A\B0) = 0.

(ii) If Be s i

then the mapping x0

f-* A of the subset (x 0

A e B n s i ) с: Еф (s i) into ( s i ,

m) is continuous by Theorem 2.2, whence B n s i is a separable subset of (s i ,

m). Now, since (iii) has been already verified, choose a sequence (B„) satisfying (iii); we may also assume that (Bn) is increasing. For each n let £ên be a countable dense subset of (B „ n s i, gM).

Let A e s i and take any e > 0. Since vM(A\ 1J B„) = 0 and A e Q c, we have

ft

vM(A\A n Bn) -

0 as n-> oo, so there is n such that vM(A\A n Bn) ^ e.

Then choose Be@}n so that

vM((A n B „ )- B ) ^ s.

Then

V„C4-B) s: + s: 2s.

Thus = U is a countable dense subset of (.^/, tju).

n

Now, conversely, suppose that conditions (i)—(iii) are satisfied. Let D be a countable dense subset of X ; using (ii) and (iii), choose B„ and rM„

as above, so that = (J 3tn is a countable dense subset of (s i, g M)- It is

n

enough to prove that {dxB’ d eD , Be@t0} is a dense subset of {

• x e X , A e s i ) . Let x e X , A e s i and take an arbitrary e > 0. Using ( + + ) it is easy to find d eD so that

N (

-<1

) ^ £•

Orlicz spaces o f vector functions 185

By ( + ) there is 3 > 0 such that N (dxE) < e if E cz A and vM(E) ^ ô. Fix an n e N so that vM(A\Bn) ^ ô; then

M(dxA-dXA\Bn) ^ e-

Next, using ( + ) again, find rj > 0 so that N (

I

) ^ e if F a A и B„ and V

(F) ^ rç. ‘ If now is chosen so that vM((A n Bn) — B) ^ q, then

N {dxAnBn-dXB) < 2e.

Gathering together these inequalities we have finally N (

X

— dxs) < 4e, which completes the proof.

4. Examples. Though perhaps not in such a great generality as treated in this paper, spaces of measurable functions (usually scalar-valued) “based on a family of measures” seem to occur quite often in analysis and prob­

ability theory. L p-spaces of such a type are used in mathematical statistics, see e.g. [14]; applications to statistics served Rosenberg [12] as a motivation for considering Orlicz spaces of a similar type. Such a generalization of Orlicz spaces has also been investigated by Musielak and Waszak [11].

We would like to indicate few other instances, where the fact that a family of measures is involved is somewhat hidden.

Let (T, <2, m) be a measure space, X the space of (real or complex) scalars and, for 1 ^ p < oo, <P(x,t) = |x|p (x e X , t e T ).

Let G be a set of measurable maps т: (T, Q )-+ (T , Q) (in particular G may be the group of all measure preserving invertible maps). For each

t

e G let

цх{А) = т( ^t г(Л)), A eQ .

Set M = (pT: r e G }. Then L p(S , Q , M ; X ) is a Lorentz type L p-space.

In particular, if T = N , Q = 2N, m{n} = wn > 0, w = (wj^y and G is the set of all permutations of N , we obtain the familiar Lorentz sequence space d(w,p), see Lindenstrauss and Tzafriri [9].

Now let F be a family of non-negative measurable functions on T.

For each f e F let

pf (A) = J fdm, A eQ .

A

Define M = {pf : f e F } . Then L P{S, Q, M ; X ) consists of those measurable functions x : T-> X such that

sup J \x(t)\p f ( t ) dm (t) < 00.

f e F T

186 L. D r e w n o w s k i and A. K a m i n s k a

The geometrical structure of this space may be quite complicated. Thus, for example, the Banach lattice without the approximation property constructed by Szankowski [13] is easily seen to be a space of this type for p = 1.

Finally, consider the following example: D = (z e C : |z| < 1}, Q = the Borel (7-algebra of D, and let, for 0 < r < l,m r be the normalized rotation invariant Borel measure on Ur — {z e C : |z| = r}. Extend each mr to Q by setting

pr (A) = mr (A n Ur), A

Q , and let M = {pr: 0 < r < 1}.

Then the space L p(D, Q, M ; C) consists of those measurable functions x : D -> C which satisfy

2

sup j \x(relt)\r dt < oo.

0 < r < i о

Clearly, it contains the classical Hardy space Hp as a closed subspace (cf. Duren [4]).

References

[1 ] L. D r e w n o w s k i, О pewnych zagadnieniach z teorii funkcji zbioru i funkcji calkowalnych, Thesis, Poznan 1970.

[2 ] —, Topological rings o f sets, continuous set functions integration 1, Bull. Acad. Polon.

Sci. Sér. Sci. Math. Astronom. Phys. 20 (1972), p. 269-286.

[3 ] N. D u n fo r d and J. T. S c h w a r tz , Linear operators, Part I, New York 1958.

[4 ] P. L. D u ren , Theory o f H p spaces,N ew Y ork 1970.

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

[6 ] A. K a m in s k a , О pewnych uogôlnionych przestrzeniach Orlicza funkcji wektorowych,Thesis, Poznan 1978.

[7 ] A. К o z ek , Orlicz spaces o f functions with values in Banach spaces, Comment. Math.

19 (1976), p. 259-288.

[8 ] —, Convex integral functionals on Orlicz spaces, ibidem 21 (1979), p. 109-135.

[9 ] J. L in d e n s tr a u s s and T z a f r i r i , Classical Banach spaces I, Springer-Verlag, Berlin- Heidelberg 1977.

[1 0 ] J. M u s ie la k and W. O r lic z , On modular spaces, Studia Math. 18 (1959), p. 49-65.

[11] J. M u s ie la k and A. W a s z a k , Countably modulared spaces connected with equisplittable families o f measures, Comment. Math. 13 (1970), p. 267-274.

[12] R. L. R o s e n b e r g , Orlicz spaces based on families o f measures, Studia Math. 35 (1970), p. 16-49.

[1 3 ] A. S z a n k o w s k i, A Banach lattice without the approximation property, Israel J. Math.

24 (1976), p. 329-337.

[14] J.-R. B a rra , Notions fundamentals de statistique mathématique,Moskva 1974 (in Russian).