measurable subset of possibly non-measurable set. Let $X$ be a Polish space. Let $T$ be a possibly uncountable set of Borel probability measures on $X$.
Suppose there exists a subset $A$ of $X$ with the following property:
For each $\mu\in T$, there exists a measurable subset of $A$, $S_{\mu}\subset A$, such that $\mu(S_{\mu})=1$.
Question:Does there exist a measurable subset $S$ of $A$ such that
$\mu(S)=1$ for all $\mu\in T$?

Let $A$ be any non-measurable set and $T=\{\delta_x: x\in A\}$.
Then the hoped-for measurable set $S$ does not exist. 
