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 ?

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- March 9th 2009, 12:52 AMmanjohn12dense
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 ? - March 9th 2009, 04:38 AMOpalg
(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. - March 12th 2009, 12:54 PMmanjohn12
- March 12th 2009, 03:56 PMHallsofIvy
- March 12th 2009, 04:02 PMHallsofIvy
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. - March 12th 2009, 04:33 PMmanjohn12
- March 13th 2009, 01:45 AMOpalg