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 theorem in Banach spaces. Answer to a question of F. Topsøe is given.