WEAK VS. NORM COMPACTNESS IN L_{1}: THE BOCCE CRITERION

Maria Girardi

Studia Math. 98 (1991) 95–97 Abstract. We present a new simple proof that if a relatively weakly compact subset of L_{1} satisfies the Bocce criterion (an oscillation condition), then it is relatively norm compact. The converse of this fact is easy to verify. A direct consequence is that, for a bounded linear operator T from L_{1} into a Banach space 𝔛, T is Dunford-Pettis if and only if the subset T^{∗}(B(𝔛^{∗})) of L_{1} satisfies the Bocce criterion.

A relatively weakly compact subset of L_{1} is relatively norm compact if and only if it satisfies the Bocce criterion (an oscillation condition) [G1]. We shall present a new simple proof that if a relatively weakly compact subset of L_{1} satisfies the Bocce criterion, then it is relatively norm compact. The converse is easy to verify.

Recall that a Banach space 𝔛 has the complete continuity property (CCP) if each bounded linear operators from L_{1} into 𝔛 is Dunford-Pettis (i.e. maps weakly convergent sequences onto norm convergent sequences). The CCP is a weakening of the Radon-Nikodým property and of strong regularity. Since a bounded linear operator T from L_{1} into 𝔛 is Dunford-Pettis if and only if the subset T^{∗}(B(𝔛^{∗})) of L_{1} is relatively norm compact, the above fact gives that T is Dunford-Pettis if and only if T^{∗}(B(𝔛^{∗})) satisfies the Bocce criterion. This oscillation characterization of Dunford-Pettis operators leads to dentability and tree characterizations of the CCP [G2]. Namely, 𝔛 has the CCP if and only if all bounded subsets of 𝔛 are weak-norm-one dentable. Also, 𝔛 has the CCP if and only if no bounded separated δ-trees grow in 𝔛, or equivalently, no bounded δ-Rademacher trees grow in 𝔛.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 47B38, 46B20, 28B99.

MARIA GIRARDI 2

Throughout this note, 𝔛 denotes an arbitrary Banach space. The triple (Ω, Σ, μ) refers to the Lebesgue measure space on [0, 1], Σ^{+} to the sets in Σ with positive measure, and L_{1} to L_{1}(Ω, Σ, μ). All notation and terminology, not otherwise explained, are as in [DU].

[G1] introduces the following definitions.

Definitions. For f in L_{1} and A in Σ^{+}, the Bocce oscillation of f on A is given by

Bocce-osc f|_{A} ≡ \frac{∫_{A} |f − ^{\frac{∫_{A} f dμ}{μ(A)}}| dμ}{μ(A)} .

A subset K of L_{1} satisfies the Bocce criterion if for each ϵ > 0 and B in Σ^{+} there is a finite collection F of subset of B each with positive measure such that for each f in K there is an A in F satisfying Bocce-osc f|_{A} < ϵ .

This note’s main purpose is to present a new proof to the theorem below. The author is grateful to Michel Talagrand for his helpful discussions concerning this theorem and proof.

Theorem. If a relatively weakly compact subset of L_{1} satisfies the Bocce criterion, then it is relatively L_{1}-norm compact.

We need the following lemma which we shall verify after the proof of the Theorem.

Lemma. If a subset of L_{1} satisfies the Bocce criterion, then the translate of that set by a L_{1}-function also satisfies the Bocce criterion.

Proof of the Theorem. Assume that the relatively weakly compact subset K of L_{1} is not relatively norm compact. We shall show that K does not satisfy the Bocce criterion.

WEAK vs. NORM COMPACTNESS 3

Since K is not relatively norm compact but is relatively weakly compact, there is a sequence {f_{n}} in a translate K̃ of K satisfying

(1) {f_{n}} has no L_{1}-convergent subsequence

(2) {f_{n}} converges weakly in L_{1} to 0

(3) { | f_{n} | } converges weakly in L_{1}, say to f

(4) ∫ f dμ ≥ 4ϵ for some ϵ > 0 . Set B = [ f ≥ 3ϵ ]. Condition (4) guarantees that B ∈ Σ^{+}.

Let F be a finite collection of subsets of B, each with positive measure. Choose N such that for each A ∈ F

(5) | ∫_{A} f_{N} dμ | < ϵ μ(A) (possible by (2))

(6) | ∫_{A} f dμ − ∫_{A} | f_{N} | dμ | < ϵ μ(A) (possible by (3)). Then for each A ∈ F we have that

Bocce-osc f_{N} |_{A} ≡ \frac{∫_{A} |f_{N} − ^{\frac{∫_{A} f_{N} dμ}{μ(A)}}| dμ}{μ(A)}  ≥ \frac{∫_{A} |f_{N}| dμ}{μ(A)} − \frac{| ∫_{A} f_{N} dμ|}{μ(A)}
≥ \frac{∫_{A} f dμ − ϵμ(A)}{μ(A)} − \frac{ϵμ(A)}{μ(A)} ≥ \frac{3ϵμ(A)}{μ(A)} − ϵ − ϵ = ϵ .

Thus, K̃ does not satisfy the Bocce criterion and so K also does not satisfy the Bocce criterion.

Proof of the Lemma. Let the subset K of L_{1} satisfies the Bocce criterion and f ∈ L_{1}. We need to show that the set K +f ≡ { g+f : g ∈ K } satisfies the Bocce criterion. Towards this end, fix ϵ > 0 and B ∈ Σ^{+}. Find B_{0} ⊂ B with B_{0} ∈ Σ^{+} such that f is bounded on B_{0}.

Approximate fχ_{B_{0}} in L_{∞}-norm within \frac{ϵ}{4} by a simple function f̃. Find C ⊂ B_{0} with C ∈ Σ^{+} such that f̃ is constant on C. Since K satisfies the Bocce criterion, we can find a finite collection F of subsets corresponding to \frac{ϵ}{2} and C.

Fix g + f ∈ K + f. Find A ∈ F such that Bocce-osc g|_{A} < \frac{ϵ}{2} . Note that since

f̃ is constant on A, Bocce-osc g|_{A} = Bocce-osc (g + f̃)|_{A} . Now,

Bocce-osc (g + f)|_{A} ≤ Bocce-osc (g + f̃)|_{A} + Bocce-osc (f̃− f)|_{A}

≤ Bocce-osc g|_{A} + 2 ∥ (f̃− f) χ_{A} ∥_{L_{∞}} < ϵ .

Thus K + f satisfies the Bocce criterion.

References

[DU]. J. Diestel and J. J. Uhl, Jr., Vector Measures, Math. Surveys, no. 15, Amer. Math.

Soc., Providence, R.I., 1977.

[GGMS]. N. Ghoussoub, G. Godefroy, B. Maurey, and W. Schachermayer, Some topological and geometrical structures in Banach spaces, Mem. Amer. Math. Soc., vol. 70, no. 378, Amer. Math. Soc., Providence, R.I., 1987.

[G1]. Maria Girardi, Compactness in L_{1}, Dunford-Pettis Operators, Geometry of Banach

Spaces, (to appear in Proc. Amer. Math. Soc.). [G2]. Maria Girardi, Dentability, Trees, and Dunford-Pettis operators on L_{1}, (to appear in

Pacific J. Math.). [T]. M. Talagrand, Pettis integral and measure theory, Mem. Amer. Math. Soc., vol. 51,

no. 307, Amer. Math. Soc., Providence, R.I., 1984.

Maria Girardi, Department of Mathematics, University of Illinois, 1409 W. Green St., Urbana, IL 61801.