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

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\...

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Bibliographic Details
Published in:arXiv.org 2012-08
Main Author: Plichko, Anatolij
Format: Article
Language:English
Subjects:
Banach spaces
Continuity
Error analysis
Half spaces
Online Access:Get full text
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Summary: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. Topsøe is given.
ISSN:2331-8422