On selection theorems with decomposable values

The main result of the paper asserts that for every separable measurable space $(T,\mathfrak F,\mu)$, where $\mathfrak F$ is the $\sigma$-algebra ofmeasurable subsets of $T$ and $\mu$ is a nonatomic probability measureon $\mathfrak F$, 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 Banachspace $L_1(T,E)$ of all Bochner integrable functions $u: T\to E$, admitsa continuous selection. Our work generalizes some results of Gon\v carov andTol'stonogov.