Hello,

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

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

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

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

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

In other words, every Borel set $\displaystyle X$, can be approximated up-to an arbitrary $\displaystyle \epsilon$ 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 $\displaystyle X$ from the top with a CLOSED set, and from the bottom with an OPEN.

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

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



Any suggestions?

Thank you in advance,

Matteo