Suppose that f is a non-negative integrable function. Let for all , the set of all -algebra.

Prove that , g non-negative integrable.

Proof so far.

Now, from an earlier thread, I understand that is a measure.

Case 1) Suppose that g is a simple function, then , Ei disjoint.

Then

Which gives

Now, is this integrand equals to ? I think it is but I just can't seem to work it out right.

Thanks.