  Let $\mu$ be a probability measure on a separable Banach space $X$. A subset
$U\subset X$ is $\mu$-continuous if $\mu(\partial U)=0$. In the paper the
$\mu$-continuity and uniform $\mu$-continuity of convex bodies in $X$,
especially of balls and half-spaces, is considered. The $\mu$-continuity is
interesting for study of the Glivenko-Cantelli theorem in Banach spaces. Answer
to a question of F. Tops{\o}e is given.
