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

On the convergence sets of operator sequences on spaces of homogeneous type