Pointwise limits for sequences of orbital integrals

Claire Anantharaman-Delaroche¹

¹ Laboratoire de Mathématiques et Applications Physique Mathématique d'Orléans (MAPMO - UMR6628) Fédération Denis Poisson (FDP - FR2964) CNRS/Université d'Orléans B.P. 6759, F-45067 Orléans Cedex 2, France

Colloquium Mathematicum, Tome 118 (2010) no. 2, pp. 401-418

Résumé

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.

Détail

DOI : 10.4064/cm118-2-4

Keywords: ross stromberg published theorem about pointwise limits orbital integrals action locally compact group rho where rho right haar measure study kind problem generally actions measure space which leave sigma finite measure relatively invariant sense mit delta every where mit delta modular function consequence obtain generalization theorem civin one parameter groups measure preserving transformations original motivation circle questions treated here dates back classical problems concerning pointwise convergence riemann sums lebesgue integrable functions