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The Bourgain property and convex hulls
Author(s) -
Rodríguez José
Publication year - 2007
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200510555
Subject(s) - mathematics , banach space , pointwise , separable space , bounded function , convex hull , combinatorics , regular polygon , convex set , integrable system , linear span , space (punctuation) , property (philosophy) , discrete mathematics , pure mathematics , mathematical analysis , convex optimization , philosophy , geometry , epistemology , linguistics
Let (Ω, Σ, μ) be a complete probability space and let X be a Banach space. We consider the following problem: Given a function f : Ω → X for which there is a norming set B ⊂ B X * such that Z f,B = { x * ○ f : x * ∈ B } is uniformly integrable and has the Bourgain property, does it follow that f is Birkhoff integrable? It turns out that this question is equivalent to the following one: Given a pointwise bounded family ℋ ⊂ ℝ Ω with the Bourgain property, does its convex hull co(ℋ) have the Bourgain property? With the help of an example of D. H. Fremlin, we make clear that both questions have negative answer in general. We prove that a function f : Ω → X is scalarly measurable provided that there is a norming set B ⊂ B X * such that Z f,B has the Bourgain property. As an application we show that the first problem has positive solution in several cases, for instance: (i) when B X * is weak* separable; (ii) under Martin's axiom, for functions defined on [0, 1] with values in a Banach space with density character smaller than the continuum. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)