A Note on Translation Continuity of Probability Measures

Abstract

A probability measure on the sphere is absolutely continuous with respect to the uniform measure on the sphere if and only if the probability of any open set varies continuously as the sphere is rotated. In general, if a topological group $G$ acts transitively on a topological space $S$, and both are Hausdorff, locally compact and second countable, then a probability measure $\nu$ on the Borel sets of $S$ is absolutely continuous with respect to the unique invariant measure class on $S$ if and only if the $\nu$-probability of an open set in $S$ varies continuously under the action of the group $G$. If $S$ is a Borel $G$-space, but the action is not assumed to be transitive, then $\nu(gE)$ is continuous in $g$ for every Borel set $E$ if and only if $\nu$ is absolutely continuous with respect to a quasi-invariant measure on $S$.