
measure theory question
So I have a measurable, nonnegative 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?
rogerpodger

Hello,
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.
__________________________________________
(because )
Therefore,
But is an increasing sequence. Hence is an increasing set.
Since (because remember that ), we have
Therefore
By the monotone convergence theorem, we can conclude :
(Whew)

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.