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arXiv · 1208.6407

On uniform continuity of convex bodies with respect to measures in Banach spaces

Anatolij Plichko
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Abstract

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.

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Anatolij Plichko. 2012-08-31. On uniform continuity of convex bodies with respect to measures in Banach spaces. https://arxiv.org/abs/1208.6407

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