On selection theorems with decomposable values

Sergei M. Ageev, Dušan Repovš

DOI: http://dx.doi.org/10.12775/TMNA.2000.026

Abstract


The main result of the paper asserts that for every separable measurable
space $(T,\mathfrak F,\mu)$, where $\mathfrak F$ is the $\sigma$-algebra of
measurable subsets of $T$ and $\mu$ is a nonatomic probability measure
on $\mathfrak F$, every Banach space $E$ and every paracompact space $X$, each
dispersible closed-valued mapping $F: x \rightsquigarrow L_1(T,E)$ of $X$ into the Banach
space $L_1(T,E)$ of all Bochner integrable functions $u: T\to E$, admits
a continuous selection. Our work generalizes some results of Gon\v carov and
Tol'stonogov.

Keywords


Multivalued mapping; continuous selection; decomposable value; Banach space; nonatomic probability measure; approximate partition; nerve of covering

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