If f is measurable function such that for all measurable sets ,show that f =0 almost everywhere on .
For n=1,2,3,..., let . Then is a measurable subset of A, and so (where denotes the measure). But . Therefore . Since this holds for all n, it follows that , which implies that f is nonpositive almost everywhere. Now do the same for the sets on which , to conclude that f is nonnegative almost everywhere. Hence f=0 almost everywhere.