Let $X$ be a Banach space and $E$ a bounded set in $X$. For $x\in X,$ we set 
$M(x,E)=\sup \{\left\| x-y\right\| :y\in E\}.$ The set $E$ is called remotal
if for any $x\in X$ there exists $z\in E$ such that $M(x,E)=\left\|
x-z\right\| .$ In this paper, we prove:

$(i)$ $M(f,$ $L^{1}(I,E))=$ $\underset{I}{\int }M(f(t),$ $E)dt,$ for $f\in
L^{1}(I,$ $X).$

$(ii)$ If $E$ is closed and span$(E)$ is a finite dimensional subspace of $%
X, $ then $L^{1}(I,$ $E)$ is remotal in $L^{1}(I,$ $X)$. Some other results are presented.