  In 1967, Ross and Str\"omberg published a theorem about pointwise limits of
orbital integrals for the left action of a locally compact group $G$ onto
$(G,\rho)$, where $\rho$ is the right Haar measure. In this paper, we study the
same kind of problem, but more generally for left actions of $G$ onto any
measured space $(X,\mu)$, which leaves the $\sigma$-finite measure $\mu$
relatively invariant, in the sense that $s\mu = \Delta(s)\mu$ for every $s\in
G$, where $\Delta$ is the modular function of $G$. As a consequence, we also
obtain a generalization of a theorem of Civin, relative to 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 relative to Lebesgue integrable functions.
