measure theory question
So I have a measurable, non-negative function f. I am to show that the integral of f with respect to x is zero if and only if the set of xsthat make f(x) > 0 is a set of measure zero. To show the first half of this implication, I am using a contrapositive argument. My classmate suggested that I use simple functions to approximate the integral below, but I'm not sure how to do this. Any thoughts?
Here is a proof..
Let be an increasing positive sequence that converges to , and where is a simple function.
We can then say that
Note that (this is very simple to prove with the above formula of )
Also note that saying is equivalent to saying since we work on positive functions.
But is an increasing sequence. Hence is an increasing set.
Since (because remember that ), we have
By the monotone convergence theorem, we can conclude :
Here's another proof for the more difficult implication (a more direct one, without simple functions):
Suppose , where is is measurable.
Assume by contradiction that .
Choose a sequence strictly decreasing to 0. Since , where the sets in the union form an increasing sequence, you have: . As a consequence, there exists such that , which implies: , in contradiction with the initial assumption.