Let be a dense subset of . Suppose that is integrable on and for all . Prove that .
So every point in is either is in or a limit point of . So define .
From here how would we show that ?
(This result is false for the Lebesgue integral, so I'll assume that it refers to the Riemann integral.)
Show that, for any dissection of the interval [a,b], the lower Riemann sum for f is ≤0 and the upper Riemann sum is ≥0. It follows that if f is integrable then its integral is 0.
That is the "lower Riemann sum" Opalg is talking about! In any interval of any partition, you can always choose an so that so you can always have . Now, because you are given that the function is integrable, the limit, using any finer and finer partitions must be the same.