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For a non-mathematician (physicist) all the integrals and definitions are equal so what are the differences among:

  • The Lebesgue integral

  • The Riemann Integral

  • The Riemann-Stiejles integral

Why aren't all the same?

Winther
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Jose Garcia
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  • Related: Lebesgue integral basics – Winther Dec 17 '16 at 12:14
  • For a physicist all functions are continuous (or left continuous, or right continuous) at all but finitely many points of singularity. Well, if you consider only those functions, then integrals are just all the same. Differences of these types of integrals occur when you consider "bad" functions. – Crostul Dec 17 '16 at 12:24
  • This is a nice exposition of some of the differences between the Lebesgue and Riemann integrals. It might also be worth noting that there are other theories of integration: Henstock-Kurzweil, Choquet, etc. Admittedly, alternative definitions are tend to just be generalisations, but so are Riemann-Stieltjes and Lebesgue, in a sense. – Theoretical Economist Dec 17 '16 at 13:06
  • http://matheducators.stackexchange.com/questions/98/comparison-of-different-concepts-of-integral – kjetil b halvorsen Dec 17 '16 at 16:57
  • Related: http://math.stackexchange.com/questions/1497662/how-much-do-we-really-care-about-riemann-integration-compared-to-lebesgue-integr http://math.stackexchange.com/questions/53121/how-do-people-apply-the-lebesgue-integration-theory – leonbloy Dec 17 '16 at 17:38

1 Answers1

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First of all, the definitions are not same, not even nearly same. The Riemann-Stieltjes integral is a generalized version of the Riemann integral, which is a basic foundation and is useful for many practical situations. But there are still functions which are not integrable under Riemann's definition or even in the modified setup established by Stieltjes. The Lebesgue integral supersedes the previous integrals in the sense that it deals with a lot of functions that are not Riemann/Riemann-Stieltjes integrable, but the two integrals coincides when they both exist.

Josh O'Brien
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    With some small caveats: Riemann integrable does imply Lebesgue integrable, but there are some improper Riemann integrals which are not Lebesgue integrable. – Winther Dec 17 '16 at 12:18
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    This answer might suggest that the point of the Lebesgue integral is just to extend the amount of integrable functions. It isn't https://en.wikipedia.org/wiki/Lebesgue_integration#Limitations_of_the_Riemann_integral – leonbloy Dec 17 '16 at 17:34