What kinds of distributions $\mu$ on $\mathbb R^n$ (seen as a measurable space, equipped with Borel $\sigma$-algebra) have the property that every Borel subset can be approximated arbitrarily well (in terms of meausure deficit) by one of its compact subsets ? That is, such that
For every $\epsilon>0$ and every Borel $A$ with $\mu(A)>0$, there exists a compact set $B \subseteq A$ such that $\mu(B) \le \mu(A)$ and $\mu(B) \ge \mu(A) - \epsilon$.
My wild guess is that any $\mu$ which has density w.r.t Lebesgue measure on $\mathbb R^n$ should have this property, but I'm not quite sure.