Lebesgue integral question:
Consider measure space with a summable function.
I want to show that if then a.e. and therefore a.e.
My proposed proof is as follows:
Assume that for some .
Since is positive measurable function it follows that it is the limit of an increasing sequence of simple functions which I will show as :
Using Beppoi Levi I know that the limit of the integral is the integral of the limit which gives the following:
This is only possible if for any we have .
for all and it follows that either or .
Since we assumed that is nonzero for some it follows that there is some where therefore there must be some such that and which gives for .
It follows that a.e. and therefore a.e.
Is this proof fine? Thanks for assistance.