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.