Well known theorem of Ulam says, that each probability measure $\mu$ defined on Borel subsets of polish space $X$ satisfies the following condition: for each $\epsilon>0$ there is a compact subset $K$ of $X$, such that $\mu(K)>1-\epsilon$.
I wonder there are any reasonable condition on measure $\mu$ which would guarantee that for each $\epsilon>0$ there is an open set subset $U$ of $X$, such that $\mbox{cl}\,U$ is compact set and $\mu(\mbox{cl}\,U)>1-\epsilon$. Any idea? It would be very helpful for me.