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Who first realized that it is possible to define the integral of a function as the limit of the integrals of a sequence of step functions that converge uniformly to the given function?

This is normally how the Lebesgue integral is taught now (although the uniform convergence is avoided by Egorov's theorem), but Lebesgue defined the integral in an analogous way to how Cauchy, Riemann and Darboux did (although he sort of flipped everyting horizontally).

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    $\begingroup$ I believe this is essentially due to F. Riesz, but it was likely refined by others, and I don't know enough of the details to fully answer. See The Uniqueness of a Riesz-type Definition of the Lebesgue Integral by Lee Peng-Yee (1970) AND A proof of the identity of the Riesz integral and the Lebesgue integral by Whyburn (1931) AND a google search for Ettlinger + "horizontal functions". $\endgroup$ – Dave L Renfro Jan 18 at 19:56
  • $\begingroup$ It is not true that uniform limits of step functions give you all Lebesgue integrable functions, even not all bounded ones. $\endgroup$ – Alexandre Eremenko Jan 20 at 14:47
  • $\begingroup$ @AlexandreEremenko I've been away from function theory too long. Can you give an example of an integrable function which cannot be expressed as the limit of step functions (or for which there is no set of step functions which converge to said function)? Thanks in advance $\endgroup$ – Carl Witthoft Jan 21 at 13:11
  • $\begingroup$ @Carl Witthoft: Dirichlet function (zero at irrationals, one at rationals) is evidently not a uniform limit of step functions. The uniform distance from any step function is $\geq 1$. $\endgroup$ – Alexandre Eremenko Jan 21 at 13:58
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    $\begingroup$ Maybe we should be clear that "step function" means the indicator function of an interval. It's a confusing point for learners of measure theory when we introduce approximating integrable functions by simple functions by drawing a picture of linear combinations of step functions approximating some function, even though we will need to use indicator functions of complicated Borel sets in the general case (so linear combinations of step functions make very atypical simple functions). $\endgroup$ – Robert Furber Jan 21 at 19:08

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