Pointwise Convergence of Sequences of Singular Measures

Abstract

We investigate the almost everywhere convergence of sequences of convolution operators given by probability measures ${\mu }_n$ on $\mathbb{R}$. If this sequence of operators constitutes an approximate identity on a particular class of functions $\mathcal{F}$, under what additional conditions do we have ${\mu }_n\ast f\to f$ a.e. for all $f\in \mathcal{F}$? We focus on the particular case of a sequence of contractions $C_{t_n}\mu$ of a single probability measure $\mu$, with $t_n\to 0$, so that that the sequence of operators is an approximate identity.