Remarks on Chacon’s biting lemma. (English) Zbl 0678.46023

In the abstract of this paper the authors indicate that a new approach is provided for the proof of ‘Chacon’s biting lemma’, which is stated in terms of Banach space-valued functions.
Let (\(\Omega\),\({\mathcal F},\mu)\) be a measure space with finite positive measure, and let \((X,\| \cdot \|_ X)\) be a Banach space. Then the main result of the paper states that if \(\{f_ j\}\) is a bounded sequence in \(L^ 1(\Omega,X)\), so that \(\sup_{j\geq 1}\int \| f_ j\|_ X d\mu =C_ 0<\infty\), then there is a function f in \(L^ 1(\Omega,X)\), a subsequence \(\{f_{k_ j}\}\) of \(\{f_ j\}\) and a non- increasing sequence \(\{E_ j\}\) of sets in \({\mathcal J}\) with \(\lim_{j\to \infty}\mu (E_ j)=0\), such that, for \(m=1,2,3,...\), \(f_{k_ j}\to f\) (as \(j\to \infty)\) weakly in \(L^ 1(\Omega \setminus E_ m,X)\).
Reviewer: G.O.Okikiolu


46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
49J45 Methods involving semicontinuity and convergence; relaxation
46G10 Vector-valued measures and integration
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