  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.
