Fix . Take any Then belongs to exactly one of or . If , for example, this means that . But . Hence .
This is my first post on this site, so please excuse me if the post is a bit rough. I'm currently reading Rudin's Real and Complex Book and ran across Theorem 1.17
Let be measurable. There exist simple measurable functions on X such that.
For and for , define
and and put where are the characteristic functions of E and F, respectively.
Part a) So we know that E and F are measurable sets because the pull back of a Borel set through a measurable function is a measurable set. And with a little algebra we can see that How do you prove that the simple functions are less than or equal to f for all n? I can do this with finite examples, but don't know how it to extend it to an arbitrary function.
Part b) If then and we're good. If then we should be able to show that for large enough n. I am having problems seeing how to actually arrive at this result.
Any help would be appreciated,