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"Lebesgue's Theorem" states that for any bounded $f:[a,b] \to \mathbb{R}$, $f$ is Riemann Integrable iff $m\{x:f \text{ is not continuous at x }\}=0$, and if so Riemann's integral coincides with Lebesgue's. ($m$ is Lebesgue's measure).

Does there exist a generalization of this theorem for higher dimensions? Can I have a proof or a reference please?

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up vote 1 down vote accepted

First note that the Lebesgue-Riemann theorem states that $f:[a,b]\to \mathbb R$ is Riemann integrable if, and only if, $f$ is bounded and is continuous almost everywhere (if a function is not bounded, it can't be Riemann integrable).

A proof of the more general result for multivariable functions can be found in volume 1 of "Real Analysis" by Duistermaat. A better exercise in fact will be to take this wiki proof and adapt it to $\mathbb R^n$.

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Shouldn't "almost always" be "almost everywhere"? – Pedro Tamaroff Mar 7 '13 at 22:11
@PeterTamaroff corrected, thanks. – Ittay Weiss Mar 7 '13 at 22:18
Could you please state the generalization? – user1337 Mar 7 '13 at 23:07

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