Something connected with Ulam's tightness theorem 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.

Think about Brownian Motion on ${\mathcal C}[0,1]$ that starts at 0. This is a probability measure on the polish space ${\mathcal C}[0,1]$. This is also a Banach space that is infinite dimensional.  Such a space cannot have a precompact open subset -- normed linear spaces with this property are finite-dimensional. Therefore, I think you are out of luck here.
