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Is there also a name associated to the Lebesgue dominated convergence theorem like Beppo-Levi or Fatou? Would Lebesgue be reasonable? Who did originally prove it?

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    $\begingroup$ @Clarinetist: Is this question rhetoric? $\endgroup$ Oct 18, 2014 at 13:28
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    $\begingroup$ I believe that it likely is attributable to Lebesgue, and the Wikipedia article seems to do so implicitly with the slight change in nomenclature: Lebesgue's dominated convergence theorem. However let me see if I can find something in the way of a definitive reference. $\endgroup$
    – hardmath
    Oct 18, 2014 at 13:38

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The result is due to Lebesgue. Bogachev (Measure Theory, Volume 1, Springer, page 428) writes:

"The Lebesgue dominated convergence theorem in the general case (with an integegrable majorant) was given by him [Lebesgue] [...]"

The specific reference that is given is (the link leads to a free copy):

H. Lebesgue, Sur la méthode de M. Goursat pour la résolution de l'équation de Fredholm, Bulletin de la Société Mathématique de France, 36 (1908), 3-19.

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  • $\begingroup$ The dominated convergence result may have appeared earlier, in Lebesgue's thesis (published 1902). I must trudge to the library, it seems, to find a copy of Hawkin's Lebesgue's Theory of Integration for further details. $\endgroup$
    – hardmath
    Oct 18, 2014 at 15:25
  • $\begingroup$ There were weaker versions earlier, as implied by the formulation "in the general case." Specifically, Lebesgue's "Leçons sur l'integration" (1904) already contains a result for $f_n$ uniformly bounded (and something might well be in his thesis already); in the paper I link to earlier results of the form are mentioned in passing when the result is formulated (page 12). But, I do not know the history in full detail. However, I interpreted the question as asking for what typically goes now as Dominated Convergence theorem, which is in the paper I link to, but not yet in his "Leçons". $\endgroup$
    – quid
    Oct 18, 2014 at 16:00

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