Let be a Borel measure on .

Denote as the domain of

Then for any , we have:

By a theorem, I also know that:

Question: If , and if , prove that where is a countable union of open subset and

Proof so far.

Case 1. If :

Given , there exists open sets with , since is the infimum of these sets.

Define .

Then we know the followings:

i. , implies that

ii.

implies that

Using both i and ii, I then conclude that , in which implies , implies that with

Case 2. :

In here, I want to do the same trick. So can I assume that ? I know I can do that for any semi-finite measures.

Any hints? Thank you.