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I am self-studying product measures, and the book I am reading did not address this question about the relationship between the integrability with respect to product measure and the integrability of each section, which I think is interesting and confusing. To be more specific:

Let $(X,\mathscr{A},\mu)$ and $(Y,\mathscr{B},\nu)$ be $\sigma$-finite measure space. Let $f:X\times Y\to[-\infty,+\infty]$ be $\mathscr{A}\times\mathscr{B}$-measurable. Define the sections $f_x:Y\to[-\infty,+\infty]$ and $f^y:X\to[-\infty,+\infty]$ by letting $f_x(y)=f(x,y)$ and $f^y(x)=f(x,y)$. Then is it true that $f$ is $\mu\times\nu$-integrable if and only if $f_x$ is $\nu$-integrable and $f^y$ is $\mu$-integrable?

I couldn't find a good way to prove or disprove it. Could someone please help me out? Thanks a lot in advance!

Beerus
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    https://math.stackexchange.com/questions/647235/counterexample-to-measurable-in-each-variable-separately-implies-measurable – Andrew Jan 04 '25 at 19:38
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    If $f$ is $\mu\times\nu$-integrable, then $f_x$ is $\nu$-integrable and $f^y$ is $\mu$-integrable by Fubini's theorem. I'm not entirely sure about the other direction, though. – user408858 Jan 04 '25 at 20:13
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    I believe you can find a function $f$, such that the integrals $\int f_x d\nu$ and $\int f^y d\mu$ do not coincide. If I'm not mistaken, this implies that $f$ is not $\mu\times\nu$-integrable. – user408858 Jan 04 '25 at 20:18
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    This turns to some set-theoretic issues. The relevant keyword is "strong Fubini theorems." – Michael Greinecker Jan 05 '25 at 01:44
  • @MichaelGreinecker Could you please provide some references, like texts, that cover strong Fubini theorems? Thank you very much! – Beerus Jan 05 '25 at 03:44
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    https://www.ams.org/journals/tran/1990-321-02/S0002-9947-1990-1025758-0/ – Michael Greinecker Jan 05 '25 at 07:26

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