## Approximation of borel sets from the top with closed sets.

Hello,

I have this problem that I don't know how to approach.

Suppose we have a topological space $T$.
Consider the Borel $\sigma$-algebra $\mathbb{B}$, generated by the open of T.

Fix a probability measure $(T,\mathbb{B},\mu)$

It is known that for every $X\in \mathbb{B}$, and for every $\epsilon\in[0,1]$, there exist an open set $O$ and a closed set $C$, $C\subseteq X\subseteq O$ such that

• $\mu(O) \leq \mu(X)+\epsilon$
• $\mu(C) \geq \mu(X)-\epsilon$

In other words, every Borel set $X$, can be approximated up-to an arbitrary $\epsilon$ from the top (with an open set) and from the bottom (with a closed set).

What I'm trying to understand if also the contrary is possible, i.e. it is also possible to approximate any $X$ from the top with a CLOSED set, and from the bottom with an OPEN.

More formally I wonder if for every $X\in \mathbb{B}$, and for every $\epsilon\in[0,1]$, there exist an open set $O$ and a closed set $C$, $O\subseteq X\subseteq C$ such that

• $\mu(O) \geq \mu(X)-\epsilon$
• $\mu(C) \leq \mu(X)+\epsilon$

Any suggestions?