Hello all I was presented with this question from Folland's real analysis second edition on Radon measures which I am stuck on and so would really appreciate the help on.

I m a novice in Radon measures especially in the concepts and abstractions so any help would be appreciated. It is problem #7 on page 220

It reads as follows:

Just in case, the definition of Radon Measure used in the book is:

A Radon measure on X is a Borel measure that is finite on all

compact sets, outer regular on all Borel sets, and inner regular on all open sets.

I also know for a fact that Radon measures are also inner regular on all their sigma finite sets.

I have tried to attack the problem many times but to no avail as for some reason I cannot seem to incorporate the given assumption that X is sigma finite.

Also the assumption is X is locally compact Hausdorff space.

I would really appreciate the help Thanks