### On selection theorems with decomposable values

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.

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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