Pointwise Convergence of Sequences of Singular Measures

We investigate the almost everywhere convergence of sequences of convolution operators given by probability measures μₙ 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 μₙ f → f a.e. for all f ∈ F? We focus on the particular case of a sequence of contractions Ctₙμ of a single probability measure μ, with tₙ → 0, so that that the sequence of operators is an approximate identity.