You should simply write , and use linearity of integral like you did before.
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.
Now, is this integrand equals to ? I think it is but I just can't seem to work it out right.