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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

##### MSC:
 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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