A characterization of almost everywhere continuous functions

Abstract

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 we will show that a function \(f\) is almost everywhere continuous with respect to \(\mu\in{\mathcal M}(X)\) if and only if \(\lim_{n\to\infty} \int_{X}f\,d\mu_n=\int_{X}f\,d\mu\), for all sequences \(\{\mu_n\}\) in \({\mathcal M}(X)\) such that \(\mu_n\) converges weakly to \(\mu\).