Hello,

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

Suppose we have a topological space .

Consider the Borel -algebra , generated by the open of T.

Fix a probability measure

It is known that for every , and for every , there exist an open set and a closed set , such that

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

More formally I wonder if for every , and for every , there exist an open set and a closed set , such that

Any suggestions?

Thank you in advance,

Matteo