- #1
psie
- 282
- 33
- TL;DR Summary
- I'm stuck with a small detail in the proposition below. I assume the reader is familiar with the space
, the functions of bounded variation. We also have , where stands for normalized; this is the space of functions which are right continuous and .
Let be Lebesgue measure. It is another proposition that the functions are in one-to-one correspondence between complex Borel measures, e.g. induces a complex measure such that . Then in Folland's real analysis text,
I'll omit the direction. In the direction, we suppose that for a Borel set and we want to show . If and are as in the definition of absolute continuity of (see here), we can find open sets such that . This is possible by so-called regularity of . To be exact, we have , and then by definition of infimum we can find such a sequence. However, it is also claimed that . Why is this true?
Some observations; is a complex measure which is regular. This means the positive measure is regular. So for some sequence of open sets. But it is not clear to me that we can simply choose (and why we'd get convergence for rather than ).
3.32 Proposition. If, then is absolutely continuous iff .
I'll omit the
Some observations;