Home | 18.01 | Chapter 12 | Section 12.4

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This definition is actually superior to the Riemann definition, because it exists even for functions which fluctuate horribly. Thus, the Lebesgue integral of the function that is 1 on the rational numbers and 0 on irrationals, is 0 for any interval. The Riemann integral of this highly discontinuous function does not exist.