On certain regularity properties of Haar-null sets
Let $X$ be an abelian Polish group. For every analytic Haar-null set $A\subseteq X$ let $T(A)$ be the set of test measures of $A$. We show that $T(A)$ is always dense and co-analytic in $P(X)$. We prove that if $A$ is compact then $T(A)$ is $G_\delta $ dense, while if $A$ is non-meager then $T(A)$ is meager. We also strengthen a result of Solecki and we show that for every analytic Haar-null set $A$, there exists a Borel Haar-null set $B\supseteq A$ such that $T(A)\setminus T(B)$ is meager. Finally, under Martin's Axiom and the negation of Continuum Hypothesis, some results concerning co-analytic sets are derived.