  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.
