Weakly precompact subsets of L₁(μ,X)

Abstract

Let (Ω, Σ, µ) be a probability space, X a Banach space, and L1(µ, X) the Banach space of Bochner integrable functions f : Ω → X. Let W = {f ∈ L1(µ, X) : for a.e. ω ∈ Ω, f (ω) ≤ 1}. In this paper we characterize the weakly precompact subsets of L1(µ, X). We prove that a bounded subset A of L1(µ, X) is weakly precompact if and only if A is uniformly integrable and for any sequence (fn) in A, there exists a sequence (gn) with gn ∈ co{fi : i ≥ n} for each n such that for a.e. ω ∈ Ω, the sequence (gn(ω)) is weakly Cauchy in X. We also prove that if A is a bounded subset of L1(µ, X), then A is weakly precompact if and only if for every > 0, there exist a positive integer N and a weakly precompact subset H of N W such that A ⊆ H + B(0), where B(0) is the unit ball of L1(µ, X).

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