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

Let μ be a probability measure on a separable Banach space X. A subset U⊂X is μ-continuous if μ(∂U)=0. In the paper the μ-continuity and uniform μ-continuity of convex bodies in X, especially of balls and half-spaces, is considered. The μ-continuity is interesting for study of the Glivenko–Cantelli...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 2013-05, Vol.401 (1), p.349-356
Main Author: Plichko, Anatolij
Format: Article
Language:English
Subjects:
[formula omitted]-continuous set
Convex body
Glivenko–Cantelli theorem
Ideal
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Summary:Let μ be a probability measure on a separable Banach space X. A subset U⊂X is μ-continuous if μ(∂U)=0. In the paper the μ-continuity and uniform μ-continuity of convex bodies in X, especially of balls and half-spaces, is considered. The μ-continuity is interesting for study of the Glivenko–Cantelli theorem in Banach spaces. Answer to a question of F. Topsøe is given.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2012.11.039