On the Glivenko-Cantelli Theorem

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

“…In connection with Corollary 2.6, it is interesting to localize natural measures µ and uniformly µ-continuous subclasses U ⊂ H. Two such examples were presented in [11] and [8]; we will present one more. The following statement shows a relation between the uniform µ-continuity of balls and the uniform µ-continuity of half-spaces whose boundaries are tangent to these balls.…”

“…Uniform µ-continuity was repeatedly used without a name (see e.g. [11], [3, p. 2], [13, p. 282], [15, p. 151]), and is equivalent to so-called µ-uniformity, which we do not consider. Of course, every uniformly µ-continuous class is µ-continuous.…”

“…[14,Theorem 2]). No infinite-dimensional Banach space is H-ideal (it follows from [11]). We obtain this result from a simple and mostly known Proposition 1.2.…”

“…These notions have their origin in the theory of empirical distributions and are connected with generalizations of the Glivenko-Cantelli theorem to metric spaces [11], [3], [13], [14], [15], [16], [12], [8]. Sometimes (see e.g.…”

“…We know only papers [11], [15], [16], [1] concerning µ-continuity directly. Let us recall some results of the mentioned papers (mainly of [15]), simultaneously presenting relevant statements of our note.…”