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Fubini's theorem relates the double integral of a function $f(x,y)$ to an iterated integral with respect to $x$ and $y$. The basic idea of this theorem for Riemann integrals of continuous functions was known for centuries, but Lebesgue proved it for bounded Lebesgue measurable functions, then Fubini proved it for Lebesgue integrable functions, then Tonelli fixed a flaw in Fubini's proof and proved it in a bit more generality.

The results of Lebesgue, Fubini, and Tonelli all pertain to Lebesgue measure. Yet today we commonly see in textbooks a Fubini-Tonelli theorem for product measures on products of abstract measure spaces. My question is, who is the first person who proved this result for abstract measure spaces? And what technique(a) did they use to prove it?

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According to Saks (1937, p. 88):

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    $\begingroup$ So is it Ulam? It's a little too terse, maybe add something about "techniques". $\endgroup$ – Conifold Nov 15 '18 at 22:29
  • $\begingroup$ Do you know what techniques any of them used? And are any of them available in English? What is the oldest English-language proof (where either the original was in English or it was later translated into English)? $\endgroup$ – user7582 Nov 20 '18 at 6:01

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