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?

Thank you in advance,

Matteo