Let $m$ be a Borel measure on a topological space $(X,\tau)$. Then $m$ is of property:
P1 if there is a sequence $\mathcal F$ of closed subsets of $X$ such that for any $\epsilon >0$ and each $A\in\tau$, there is $F\in\mathcal F$ such that $F\subseteq A$ and $m(A-F)<\epsilon$.
P2 if there is a sequence $\mathcal K$ of compact subsets of $X$ such that for any $\epsilon >0$ and each $A\in\tau$, there is $K\in\mathcal K$ such that $K\subseteq A$ and $m(A-K)<\epsilon$.
An easy result: if $X$ is compact, then P1 $\implies$ P2, but not conversely.
Q) Is there is any counterexample for this? I was not able to think of any non Hausdorff topological measure space that has P2 but not P1.
I welcome any idea, comment, or even reference.