## Real Analysis Exchange

### A characterization of almost everywhere continuous functions

Fernando Mazzone

#### 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 $\mu$.

#### Article information

Source
Real Anal. Exchange, Volume 21, Number 1 (1995), 317-319.

Dates
First available in Project Euclid: 3 July 2012