Pointwise limits for sequences of orbital integrals

TL;DR

This paper generalizes classical results on pointwise limits of orbital integrals from group actions on groups to more general measured spaces, extending the understanding of convergence in harmonic analysis.

Contribution

It extends the theorem on pointwise limits of orbital integrals to actions on measured spaces with relatively invariant measures, generalizing previous results.

Findings

Generalized Ross and Strömberg's theorem to measured spaces

Connected results to Civin's theorem on measure-preserving transformations

Provided insights into pointwise convergence of Riemann sums

Abstract

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.