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 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 ϵ >0, there exist a positive integer N and a weakly precompact subset H of NW such that A ⊆ H + ϵB(0), where B(0) is the unit ball of L₁(μ,X).
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm129-1-10,
author = {Ioana Ghenciu},
title = {Weakly precompact subsets of L1(m,X)},
journal = {Colloquium Mathematicae},
volume = {126},
year = {2012},
pages = {133-143},
zbl = {1272.46030},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm129-1-10}
}
Ioana Ghenciu. Weakly precompact subsets of L₁(μ,X). Colloquium Mathematicae, Tome 126 (2012) pp. 133-143. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm129-1-10/