CONDITIONAL WEAK COMPACTNESS IN VECTOR-VALUED ...

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 129, Number 10, Pages 2947–2953 S 0002-9939(01)06064-6 Article electronically published on April 17, 2001

CONDITIONAL WEAK COMPACTNESS IN VECTOR-VALUED FUNCTION SPACES MARIAN NOWAK (Communicated by Dale Alspach) Abstract. Let E be an ideal of L0 over a σ-finite measure space (Ω, Σ, µ) othe dual of E with supp E 0 = Ω. Let (X, k · kX ) be and let E 0 be the K¨ a real Banach space, and X ∗ the topological dual of X. Let E(X) be a subspace of the space L0 (X) of equivalence classes of strongly measurable functions f : Ω → X and consisting of all those f ∈ L0 (X) for which the scalar function kf (·)kX belongs to E. For a subset H of E(X) for which the set {kf (·)kX : f ∈ H} is σ(E, E 0 )-bounded the following statement is equivalent the set {kf (·)kX : f ∈ H} is to conditional σ(E(X), E 0 (X ∗ ))-compactness: R conditionally σ(E, E 0 )-compact and { A f (ω)dµ : f ∈ H} is a conditionally weakly compact subset of X for each A ∈ Σ, µ(A) < ∞ with χA ∈ E 0 . Applications to Orlicz-Bochner spaces are given.

1. Introduction and preliminaries Given a dual pair hL, Ki, a subset A of L is said to be conditionally σ(L, K)compact whenever each sequence in A contains a σ(L, K)-Cauchy subsequence (cf. [MN, p. 100]). The problem of characterizing relatively sequentially σ(Lp (X), Lq (X ∗ ))-compact subsets of Lebesgue-Bochner spaces Lp (X) (where 1 6 p < ∞ and q conjugate to p) over a finite measure space was considered by F. Bombal [B1 ] and J. Batt and W. Hiermeyer [BH, Theorem 2.1]. Moreover, F. Bombal character∗ ized relatively sequentially σ(Lϕ (X), Lϕ (X ∗ ))-compact subsets of Orlicz-Bochner spaces Lϕ (X) [B2 , Theorem 3]. C. Abott, E. Bator, R. Bilyeu and P. Lewis [ABBL] obtained the following characterization of conditionally σ(L1 (X), L∞ (X ∗ ))-compact subsets of L1 (X). Theorem 1.1 (cf. [ABBL, Theorem 2.5]). Let (Ω, Σ, µ) be a finite measure space. Then for a norm bounded subset H of L1 (X) the following statements are equivalent: (i) H is conditionally σ(L1 (X), L∞ (X ∗ ))-compact. 1 (ii) a) The subset R {kf (·)kX : f ∈ H} of L is uniformly integrable. b) The set { f (ω)dµ : f ∈ H} is conditionally weakly compact in X for each A ∈ Σ.

A

In this paper, by making use of Theorem 1.1 we characterize conditionally σ(E(X), E 0 (X ∗ ))-compact subsets of E(X), where E is an ideal of L0 over a σfinite measure space. Received by the editors July 6, 1998 and, in revised form, February 14, 2000. 2000 Mathematics Subject Classification. Primary 46B25, 46E40. Key words and phrases. Conditional weak compactness, vector valued function spaces. c

2001 American Mathematical Society

2947

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

2948

MARIAN NOWAK

Now we establish notation and terminology (see [AB], [KA]). Let (Ω, Σ, µ) be a complete σ-finite measure space and let L0 denote the space of equivalence classes of all Σ-measurable functions defined and finite a.e. on Ω. Let χA stand for the characteristic function of a set A and let N denote the set of all natural numbers. Let E be an ideal of L0 with supp E = Ω, and let E 0 stand for the K¨ othe dual of E, i.e., Z |u(ω)v(ω)|dµ < ∞ for all u ∈ E}. E 0 = {v ∈ L0 : Ω

We assume that supp E 0 = Ω. Let (X, k · kX ) be a real Banach space, and let SX and BX denote the unit sphere and the closed unit ball in X, resp. Let X ∗ stand for the Banach dual of X. By L0 (X) we denote the set of equivalence classes of all strongly Σ-measurable functions f : Ω → X. For f ∈ L0 (X) let us set fe(ω) = kf (ω)kX for ω ∈ Ω. Let E(X) = {f ∈ L0 (X) : fe ∈ E}. By σ(E(X), E 0 (X ∗ )) we will denote the weak topology on E(X)Rwith respect to the dual system hE(X), E 0 (X ∗ )i under the natural duality hf, gi = hf (ω), g(ω)idµ for Ω

f ∈ E(X), g ∈ E 0 (X ∗ ). The following characterization of conditional σ(E, E 0 )-compactness is needed. Proposition 1.2 ([N2 , Theorem 1.1]). For a σ(E, E 0 )-bounded subset A of E the following statements are equivalent: (i) A is conditionally σ(E, E 0 )-compact. (ii) For each v ∈ E 0 the subset {uv : u ∈ A} of L1 is Runiformly integrable. (iii) The functional pA on E 0 defined by pA (v) = sup |u(ω)v(ω)|dµ is an order u∈A Ω

continuous Riesz seminorm. 2. Conditionally σ(E(X), E 0 (X ∗ ))-compact sets in E(X) Let ca(Ω, Σ) stand for the Riesz space of countably additive set functions ν on T ∞ An = 0 Σ. For a sequence (An ) in Σ we write An &µ ∅ whenever An ↓ and µ n=1

(that is, An ↓ and µ(An ∩ A) → 0 for each A ∈ Σ with µ(A) < ∞). The following well-known result characterizes uniformly µ-continuous sets in ca(Ω, Σ). Lemma 2.1. For a subset K of ca(Ω, Σ)+ the following statements are equivalent: (i) K is uniformly µ-continuous (i.e., lim(sup ν(An )) = 0 as An &µ ∅). n

ν∈K

(ii) For each η > 0 there exist δ > 0 and A0 ∈ Σ with µ(A0 ) < ∞ such that ν(A) 6 η and ν(Ω \ A0 ) 6 η for all A ∈ Σ with µ(A) 6 δ and all ν ∈ K. We shall need the following technical result. Proposition 2.2. Let K be a subset of ca(Ω, Σ)+ such that each ν ∈ K is µcontinuous. Assume that K is not uniformly µ-continuous. Then there exist a pairwise disjoint sequence (Bn ) in Σ, a number ε0 > 0 and a sequence (νn ) in K such that νn (Bn ) > ε0 for all n ∈ N.


CONDITIONAL WEAK COMPACTNESS

2949

Proof. In view of Lemma 2.1 there exists ε0 > 0 such that either there exist a sequence (An ) in Σ and a sequence (νn1 ) in K such that µ(An ) → 0 and νn1 (An ) > 2ε0

(1)

or there exists a sequence (νn2 ) in K such that νn2 (Ω \ Ωn ) > 2ε0

(2)

whenever Ωn ↑ Ω and µ(Ωn ) < ∞ for n ∈ N. Assume that condition (1) holds. Then arguing as in [BL, p. 546] one can find a pairwise disjoint sequence (Bn ) in Σ and a subsequence (νk1n ) of (νn1 ) such that νk1n (Bn ) > ε0 . Let νn = νk1n for n ∈ N. Now assume that condition (2) holds. Let Cn = Ω \ Ωn for n ∈ N. Then Cn &µ ∅, so ν(Cn ) → 0 for each ν ∈ K. Let l1 = 1 and choose l2 ∈ N such that l2 > l1 , νl21 (Cl2 ) < ε0 . Then choose l3 ∈ N such that l3 > l2 and νl2 (Cl3 ) < ε0 . Continuing this process inductively we can find an increasing sequence (ln ) in N such that νl2n (Cln+1 ) < ε0 . Let Bn = Ωln+1 \ Ωln for n ∈ N. Then (Bn ) is a disjoint sequence and since Bn = Cln \ Cln+1 for n ∈ N, by making use of (2) we obtain that νl2n (Bn ) = νl2n (Cln ) − νl2n (Cln+1 ) > 2ε0 − ε0 = ε0 . Put νn = νl2n for n ∈ N. e {fe: f ∈ H}. For a subset H of E(X) let H= Now we are ready to state our main result. e of E is σ(E, E 0 )Theorem 2.3. Let H be a subset of E(X) such that the subset H bounded. Then the following statements are equivalent: (i) H is conditionally σ(E(X), E 0 (X ∗ ))-compact. 0 e (ii) a) H o E )-compact. nR is conditionally σ(E, f (ω)dµ : f ∈ H is a conditionally weakly compact subset of X for b) A

each A ∈ Σ, µ(A) < ∞ with χA ∈ E 0 . Proof. (i) ⇒ (ii) To prove that (a) holds, in view of Proposition 1.2 it is enough to show that for each 0 6 v ∈ E 0 the subset {fev : f ∈ H} of L1 is uniformly integrable. Assume on the contrary that there exists 0 6 v0 ∈ E 0 such that the set {fev0 : f ∈ R H} is not uniformly integrable. For each f ∈ H set νf (A) = fe(ω)v0 (ω)dµ for A

A ∈ Σ. Then νf is a non-negative µ-continuous countably additive set function on Σ but the family {νf : f ∈ H} is not uniformly µ-continuous. Hence in view of Proposition 2.2 there exist a pairwise disjoint sequence (Bn ) in Σ, a sequence (fn ) R fen (ω)v0 (ω)dµ > ε0 for each in H, and a number ε0 > 0 such that νfn (Bn ) = Bn

n ∈ N. Clearly v0 fn ∈ L1 (X), so in view of [Bu, Theorem 1.1.(4)] νfn (Bn ) = kχBn v0 fen kL1 = kχBn v0 fn kL1 (X) n Z o = sup hv0 (ω)fn (ω), g(ω)idµ : g ∈ L∞ (X ∗ ), kgkL∞ (X ∗ ) 6 1 . Bn

Hence one can produce a sequence (gn ) in L∞ (X ∗ ) with kgn kL∞ (X ∗ ) 6 1, χΩ\Bn gn = 0 and such that Z (1) hv0 (ω)fn (ω), gn (ω)idµ > ε0 . Bn


2950

MARIAN NOWAK

Set g0 =

∞ P

gn .

Then g0 ∈ L0 (X ∗ ) and kg0 kL∞ (X ∗ ) 6 1.

Clearly v0 g0 ∈

n=1

E 0 (X ∗ ), so χA v0 g0 ∈ E 0 (X ∗ ) for each A ∈ Σ. In view of the assumption (i) there exists a σ(E(X), E 0 (X ∗ ))-Cauchy subsequence (fkn )R of (fn ) so for each A ∈ R Σ, lim hfkn (ω), v0 (ω)g0 (ω)idµ exists. Setting µn (A) = hfkn (ω), v0 (ω)g0 (ω)idµ n A

A

for A ∈ Σ, in view of Nikodym’s convergence theorem (see [D, Chap. 7]), {µn : n ∈ N} is uniformly countably additive on Σ. Hence there exists m0 ∈ N such that for m > m0 , sup |µn (Bm )| 6 ε0 (see [D, Chap. 7, Theorem 10]). Hence for each n

m > m0 we get

Z hfkm (ω), v0 (ω)gkm (ω)idµ |µm (Bkm )| = Bk

m Z hv0 (ω)fkm (ω), gkm (ω)idµ 6 ε0 =

Bkm

which contradicts (1). This contradiction establishes that (a) holds. To show that (b) holds, take A ∈ Σ with χA ∈ E 0 , and let (fn ) be a sequence in H. Set g = χA x∗ where x∗ ∈ SX ∗ . Then g ∈ E 0 (XR ∗ ) and by assumption (i) there exists a subsequence (fkn ) of (fn ) such that lim hfkn (ω), g(ω)idµ exists. Since n nR Ω o R R ∗ hfkn (ω), g(ω)idµ = x fkn (ω)dµ , the set f (ω)dµ : f ∈ H is conditionally Ω

A

A

weakly compact in X. (ii) ⇒ (i) Let (fn ) be a sequence in H. Since supp E 0 = Ω there exists a sequence (Ωm ) in Σ such that Ωm ↑ Ω and µ(Ωm ) < ∞, χΩm ∈ E 0 for m ∈ N (see [Z, Theorem 86.2]). Setting Am = Ω \ Ωm for m ∈ N we see that Am &µ ∅. R e fen (ω)dµ = cm < ∞, because χΩ ∈ E 0 and H Given m ∈ N we have sup n Ωm

0

m

⊂ L1Ωm (X), and by assumption subset of L1Ωm . Combining this

is σ(E, E )-bounded. Hence {χΩm fn : n ∈ N} (a), {χΩm fen : n ∈ N} is a uniformly integrable observation with (b), in view of Theorem 1.1 we see that {χΩm fn : n ∈ N} is a ∗ 1 conditionally σ(L1Ωm (X), L∞ Ωm (X ))-compact subset of LΩm (X). ∗ In view of the above observation there exists a σ(L1Ω1 (X), L∞ Ω1 (X ))-Cauchy ∗ subsequence (χΩ1 fkn1 ) of (χΩ1 fn ). Next, there exists a σ(L1Ω2 (X), L∞ Ω2 (X ))-Cauchy 2 1 subsequence (χΩ2 fkn ) of (χΩ1 fkn ). It follows that the diagonal sequence (fknn ) has ∗ the property that for each m ∈ N (χΩm fknn ) is a σ(L1Ωm (X), L∞ Ωm (X ))-Cauchy sequence. Put hn = fknn for n ∈ N. Let g ∈ E 0 (X ∗ ). For n ∈ N let us put ( g(ω) if ω ∈ Ωn and kg(ω)kX ∗ 6 n, gn (ω) = 0 elsewhere. Given ε > 0 there exist m0 ∈ N and δ > 0 such that Z Z ε ε (2) and sup fen (ω) ge (ω)dµ 6 g (ω)dµ 6 fen (ω) e sup 4 4 n n Ω\Ωm0

A



for each A ∈ Σ with µ(A) 6 δ. For η =

ε 4cm0

2951

let

Bn = {ω ∈ Ωm0 : kg(ω) − gn (ω)kX ∗ > η}. It is easy to observe that Bn ↓ ∅, so µ(Bn ) → 0. Choose n0 ∈ N with n0 > m0 such that µ(Bn0 ) 6 δ. Then by (2) we get Z ε e (3) hn (ω) ge (ω)dµ 6 . sup 4 n Bn0

Hence, by (3) we have Z Z hhn (ω), g(ω) − gn0 (ω)idµ 6 Z (4)

6

Ωm0

Ωm0

6

e hn (ω) ge (ω)dµ + η

Bn0

Since

R

Z

e hn (ω)kg(ω) − gn0 (ω)kX ∗ dµ +

Bn0

Z

e hn (ω)kg(ω) − gn0 (ω)kX ∗ dµ

Z

Ωm0

e hn (ω)kg(ω) − gn0 (ω)kX ∗ dµ

Ωm0 \Bn0

ε ε ε e · cm 0 = . hn (ω)dµ 6 + 4 4cm0 2

hhn (ω), gn0 (ω)idµ −→ a for some a ∈ R, we can choose n1 ∈ N such that n

Ωm0

for n > n1

ε Z hhn (ω), gn0 (ω)idµ − a 6 . 4

(5)

Ωm0

Thus by (2), (4) and (5) for n > n1 we get Z hhn (ω), g(ω)idµ − a Ω

Z 6

Z hhn (ω), g(ω)idµ + hhn (ω), g(ω) − gn0 (ω)idµ

Ω\Ωm0

Ωm0

ε ε ε Z hhn (ω), gn0 (ω)idµ − a 6 + + = ε. + 4 2 4 Ωm0

This shows that (hn ) is a σ(E(X), E 0 (X ∗ ))-Cauchy subsequence of (fn ), so H is conditionally σ(E(X), E 0 (X ∗ ))-compact. Corollary 2.4. Assume that a Banach space X contains no isomorphic copy of `1 , e is σ(E, E 0 )-bounded. Then the following and let H be a subset of E(X) such that H statements are equivalent: (i) H is conditionally σ(E(X), E 0 (X ∗ ))-compact. e is conditionally σ(E, E 0 )-compact. (ii) H Proof. (i) ⇒ (ii) Obvious. (ii) ⇒ (i) In view of Theorem 2.3 it is enough to show that

nR

f (ω)dµ : f ∈ H

A

o

is a conditionally weakly compact subset of X for each A ∈ Σ such R that χA ∈ E 0 . In fact, let A ∈ Σ, µ(A) < ∞ with χA ∈ E 0 . Hence sup k f (ω)dµk 6 f ∈H


A

2952

MARIAN NOWAK

nR R sup fe(ω)dµ < ∞, so in view of the Rosenthal’s `1 -theorem [R] f (ω)dµ : f ∈ f ∈H A A o H is a conditionally weakly compact subset of X, as desired. Assume now that (E, k · kE ) is a Banach function space. Then the space E(X) othe-Bochner space. provided with the norm kf kE(X) := kfekE is usually called a K¨ othe dual E 0 can be defined as follows: The associated norm k · kE 0 on the K¨ n Z o 0 kvkE = sup u(ω)v(ω)dµ : u ∈ E, kukE 6 1 . Ω

e is σ(E, E 0 )-bounded whenever Clearly, for a subset H of E(X) the set H sup kf kE(X) < ∞.

f ∈H

Combining Corollary 2.4 and Proposition 1.2 we get: Corollary 2.5. Let (E, k · kE ) be a Banach function space, and assume that X contains no isomorphic copy of `1 . Then the following statements are equivalent: (i) The associated norm k · kE 0 on E 0 is order continuous. (ii) Every norm bounded set in E(X) is conditionally σ(E(X), E 0 (X ∗ ))-compact. We now apply the previous results to Orlicz spaces (see [KR], [L] for more details). By a Young function we mean a mapping ϕ : [0, ∞) → [0, ∞) that is convex, = 0, lim ϕ(t) = ∞. Let Lϕ be the Orlicz vanishes only at 0 and lim ϕ(t) t t t→0

t→∞

space R associated with ϕ and provided with ∗the Luxemburg norm kukϕ := inf{λ > 0 : ϕ(|u(ω)|/λ)dµ 6 1}. Then (Lϕ )0 = Lϕ , where ϕ∗ denotes the complementary Ω

Young function. We say that a Young function ψ increases more rapidly than another ϕ, in symbols ϕ≺ −ψ, if for each c > 0 there is d > 1 such that cϕ(t) 6 d1 ψ(dt) for all −ϕ. t > 0 (see [N1 ]). Note that ϕ satisfies the ∇2 -condition iff ϕ≺ As a consequence of Corollary 2.4 and [N1 , Theorem 2.5] we get: Corollary 2.6. Assume that X contains no isomorphic copy of `1 . Then for a norm bounded subset H of the Orlicz-Bochner space Lϕ (X) the following statements are equivalent: ∗ (i) H is conditionally σ(Lϕ (X), Lϕ (X ∗ ))-compact. (ii) There is a Young function ψ with ϕ≺ −ψ and such that H ⊂ Lψ (X) and sup kf kLψ (X) < ∞. f ∈H

Corollary 2.7. Assume that X contains no isomorphic copy of `1 and ϕ satisfies the ∇2 -condition. Then every norm bounded subset of Lϕ (X) is conditionally ∗ σ(Lϕ (X), Lϕ (X ∗ ))-compact. References C.D. Aliprantis and O. Burkinshaw, Locally solid Riesz spaces, Academic Press, New York, San Francisco, London, 1978. MR 58:12271 [ABBL] C. Abbott, E. Bator, R. Bilyeu, P. Lewis, Weak precompactness, strong boundedness, and weak complete continuity, Math. Proc. Camb. Phil. Soc., 108 (1990), 325-335. MR 92b:46047 [AB]



[BH] [B1 ] [B2 ] [BL] [Bu] [DU] [D] [KA] [KR] [L] [MN] [N1 ] [N2 ] [R] [Z]

2953

J. Batt, W. Hiermeyer, On compactness in Lp (µ, X) in the weak topology and in the topology σ(Lp (µ, X), Lq (µ, X 0 )), Math. Z., 182 (1983), 409-423. MR 84m:46039 F. Bombal, On the space Lp (µ, X), Rev. Real Acad. Cienc. Exact. Fis. Natur. Madrid, 74, no. 1 (1980), 131-135 (in Spanish). MR 82i:46042 F. Bombal, On Orlicz space of vector-valued functions, Collect. Math., 32, no. 1 (1981), 3-12 (in Spanish). MR 83a:46039 R. Bilyeu, P. Lewis, Uniform differentiability, uniform absolute continuity and the VitaliHahn-Saks theorem, Rocky Mtn. J. Math., 10, No. 3 (1980), 533-557. MR 82g:46083 A.V. Bukhvalov, On an analytic representation of operators with abstract norm, Izv. Vyss. Uceb. Zaved., 11 (1975), 21-32 (in Russian). J. Diestel, J.J. Uhl Jr., Vector measures, Math. Surveys, 15, Amer. Math. Soc., Providence, R.I., 1977. MR 56:12216 J. Diestel, Sequences and series in Banach spaces, Graduate Texts in Math., SpringerVerlag, New York, 1984. MR 85i:46020 L.V. Kantorovitch, A.V. Akilov, Functional analysis, Nauka, Moscow, 1984 (3rd ed.) (in Russian). M. Krasnoselskii, Ya. B. Rutickii, Convex functions and Orlicz spaces, P. Noordhoff Ltd, Groningen, 1961. MR 23:A4016 W. Luxemburg, Banach function spaces, Delft, 1955. MR 17:285a P. Mayer-Nieberg, Banach lattices, Springer-Verlag, Berlin, Heidelberg, New York, 1991. M. Nowak, Order continuous seminorms and weak compactness in Orlicz spaces, Collect. Math., 44 (1993), 217-236. MR 95g:46055 M. Nowak, Weak sequential compactness in non-locally convex Orlicz spaces, Bull. Pol. Acad. Sci., 46 (1998), 225-231. MR 99g:46034 H.P. Rosenthal, A characterization of Banach spaces containing `1 , Proc. Nat. Acad. Sci. U.S.A., 71 (1974), 2411-2413. MR 50:10773 A.C. Zaanem, Riesz spaces II, North Holland Pub. Comp., Amsterdam, New York, Oxford, 1983.

´ ski Pedagogical University, Pl. Slowian ´ ski 9, Institute of Mathematics, T. Kotarbin ´ ra, Poland 65–069 Zielona Go E-mail address: [email protected]
