Dichotomies of the set of test measures of a Haar-null set

Abstract

We prove that if $X$ is a Polish space and $F$ is a face of $P(X)$ with the Baire property, then $F$ is either a meager or a co-meager subset of $P(X)$. As a consequence we show that for every abelian Polish group $X$ and every analytic Haar-null set $A\subseteq X$, the set of test measures $T(A)$ of $A$ is either meager or co-meager. We characterize the non-locally-compact groups as the ones for which there exists a closed Haar-null set $F\subseteq X$ with $T(F)$ is meager. Moreover, we answer negatively a question of J. Mycielski by showing that for every non-locally-compact abelian Polish group and every $\sigma$-compact subgroup $G$ of $X$ there exists a $G$-invariant $F_\sigma$ subset of $X$ which is neither prevalent nor Haar-null.