  We consider sequences of operators $U_n:L^1(X)\to M(X)$, where $X$ is a space
of homogeneous type. Under certain conditions on the operators $U_n$ we give a
complete characterization of convergence (divergence) sets of functional
sequences $U_n(f)$, where $f\in L^p(X)$, $1\le p\le \infty$. The results are
applied to characterize convergence sets of some specific operator sequences in
classical analysis.
