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Abstract

In 1967, Ross and Stromberg published a theorem about pointwise limits of orbital integrals for the left action of a locally compact group $G$ on $(G,\rho )$, where $\rho $ is the right Haar measure. We study the same kind of problem, but more generally for left actions of $G$ on any measure space $(X,\mu )$, which leave the $\sigma $-finite measure $\mu $ relatively invariant, in the sense that $s\mu = {\mit\Delta} (s)\mu $ for every $s\in G$, where ${\mit\Delta} $ is the modular function of $G$. As a consequence, we also obtain a generalization of a theorem of Civin on one-parameter groups of measure preserving transformations. The original motivation for the circle of questions treated here dates back to classical problems concerning pointwise convergence of Riemann sums of Lebesgue integrable functions.