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.