# Weakly precompact subsets of L₁(μ,X)

Colloquium Mathematicae (2012)

• Volume: 129, Issue: 1, page 133-143
• ISSN: 0010-1354

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## Abstract

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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 $gₙ\in co{f}_{i}:i\ge n$ 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 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).

## How to cite

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Ioana Ghenciu. "Weakly precompact subsets of L₁(μ,X)." Colloquium Mathematicae 129.1 (2012): 133-143. <http://eudml.org/doc/283562>.

@article{IoanaGhenciu2012,
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 $gₙ ∈ co\{f_i: i ≥ n\}$ 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 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).},
author = {Ioana Ghenciu},
journal = {Colloquium Mathematicae},
keywords = {Bochner function spaces; weakly precompact sets; -sets},
language = {eng},
number = {1},
pages = {133-143},
title = {Weakly precompact subsets of L₁(μ,X)},
url = {http://eudml.org/doc/283562},
volume = {129},
year = {2012},
}

TY - JOUR
AU - Ioana Ghenciu
TI - Weakly precompact subsets of L₁(μ,X)
JO - Colloquium Mathematicae
PY - 2012
VL - 129
IS - 1
SP - 133
EP - 143
AB - 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 $gₙ ∈ co{f_i: i ≥ n}$ 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 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).
LA - eng
KW - Bochner function spaces; weakly precompact sets; -sets
UR - http://eudml.org/doc/283562
ER -

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