While reading Kallenberg's proof of Fubini theorem in Foundations of modern probability, I realized that he first proved Tonelli's theorem, then apply Tonelli to $f_+$ and $f_-$, the positive and negative part of a product absolute integrable function $f(x,y)$. Since both integrals are finite, they can be subtracted to obtain Fubini theorem for $f$.

I was wondering, however, if $f_-$ is product integrable with finite integral, then even $f_+$ integrates to $\infty$, can we still subtract a finite number from $\infty$ and still obtain the Fubini theorem. If not, is there a counter example where $f_-$ has finite integral but integral of its positive part is infinite, and iterated integrals of $f$ yields different values?

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    $\begingroup$ Note that in Tonelli's Theorem we require the space to be $\sigma$-finite. In Fubini's Theorem we don't need to explitly request the space to be $\sigma$-finite, because we request $f$ to be integrable (with finite integral), so the set $[f\neq 0]$ is $\sigma$-finite. If, in Fubini's Theorem, you request only that $f^-$ has finite integral and $f^+$ be integrable with possibly infinite integral, then you need to explicitly include the condition that the space $\sigma$-finite. $\endgroup$ – Ramiro Aug 31 '15 at 21:50
  • $\begingroup$ Thank you for your comment. I assume then for sigma-finite product measure, it is true that Fubini theorem can be made a little more general. $\endgroup$ – user138668 Aug 31 '15 at 22:00

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