Pointwise Convergence of Sequences of Singular Measures

We investigate the almost everywhere convergence of sequences of convolution operators given by probability measures ₙ on R. If this sequence of operators constitutes an approximate identity on a particular class of functions F, under what additional conditions do we have ₙ f f a. e. for all f F? We focus on the particular case of a sequence of contractions Cₓ䂸 of a single probability measure, with tₙ 0, so that that the sequence of operators is an approximate identity.