Bickel, P. J.; Millar, P. W. Uniform convergence of probability measures on classes of functions. (English) Zbl 0821.60002 Stat. Sin. 2, No. 1, 1-15 (1992). Summary: Let \(P_ n\), \(P\) be probabilities, and \(\mathbf F\), \({\mathbf F}^*\) be collections of real functions. Simple conditions are derived under which the simple convergence of \(\int f(x) P_ n (dx)\) to \(\int f(x) P(dx)\) for every \(f\) in \({\mathbf F}^*\) implies uniform convergence over \({\mathbf F} :\sup_{f \in F} | \int f(x) P_ n (dx) - \int f(x) P(dx) |\) converges to 0. Several examples are discussed, some historical and some new. Cited in 7 Documents MSC: 60B10 Convergence of probability measures Keywords:weak convergence of probabilities; uniform convergence of probabilities; Glivenko-Cantelli theorem; Vapnik-Cervonenkis class; convex sets PDFBibTeX XMLCite \textit{P. J. Bickel} and \textit{P. W. Millar}, Stat. Sin. 2, No. 1, 1--15 (1992; Zbl 0821.60002)