Weakly precompact subsets of L₁(μ,X)
Ioana Ghenciu (2012)
Colloquium Mathematicae
Similarity:
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 for each n such that for a.e. ω ∈ Ω, the sequence (gₙ(ω)) is weakly...