A characterization of almost everywhere continuous functions. (English) Zbl 0866.28003

Mazzone, Fernando

Real Anal. Exch. 21(1995-96), No. 1, 317-319 (1996).

Let \((X,d)\) be a separable metric space and \({\mathcal M}(X)\) the set of probability measures on the \(\sigma\)-algebra of Borel sets in \(X\). In this paper, the author proves that a bounded measurable function \(f:X\to\mathbb{R}\) is almost everywhere continuous with respect to \(\mu\in{\mathcal M}(X)\) if and only if \(\lim_{n\to\infty} \int_Xf d\mu_n=\int_Xf d\mu\) for any sequence \(\{\mu_n\}\) in \({\mathcal M}(X)\) weakly convergent to \(\mu\).

Reviewer: J.M.Ayerbe (Sevilla)

MSC:

28A20	Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
60B05	Probability measures on topological spaces
54C05	Continuous maps

Keywords:

almost everywhere continuous functions; measurable functions; probability measures; Borel sets