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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
    Commented Nov 15, 2018 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)? $\endgroup$
    – Keshav Srinivasan
    Commented Nov 20, 2018 at 6:01
  • $\begingroup$ Ulam's 1932 ICM announcement (of joint work with Łomnicki) does not mention the Fubini theorem, but rather it states that given two probability spaces $(X,\mu_X),(Y,\mu_Y)$ we can define a complete probability measure $\mu$ on $X \times Y$ such that $\mu(S \times T) = \mu_X(S) \cdot \mu_Y(T)$ when $S,T$ are measurable. The 1934 paper contains generalizations to infinite products, and no Fubini theorem, as far as I can see. $\endgroup$
    – Robert Furber
    Commented Oct 3, 2020 at 13:28
  • $\begingroup$ Hahn's 1933 paper contains the Fubini theorem for sets, rather than integrable functions, as statement XXXIX in §5 (i.e. $\mu(S) = \int_Y \mu_X(S_y) d\mu_Y$ in the notation of my previous comment, with $S \subseteq X \times Y$ measurable). $\endgroup$
    – Robert Furber
    Commented Oct 3, 2020 at 13:31
  • $\begingroup$ The zbMATH review of Feller's article says that it contains the abstract version of Fubini's theorem, but I can't access the article to verify this, as I sadly do not know anywhere that has copies of articles from the Bulletin of the Yugoslav Academy of Sciences. But if this is right, it indicates that the answer is Feller (or Hahn, depending on which statement is considered), rather than Ulam. $\endgroup$
    – Robert Furber
    Commented Oct 3, 2020 at 13:32

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