# Who first proved Fubini's theorem for abstract measure spaces?

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?

• 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. Oct 3, 2020 at 13:28
• 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). Oct 3, 2020 at 13:31