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(This question was on MSE, with no answers)

Consider two $\sigma$-finite measure spaces $X_1$ and $X_2$, and $X=X_1\times X_2$ the product measure space (a priori non-completed).

Take two functions measurable functions $f$ and $g$ on $X_1\times X_2$ that are almost everywhere equal (meaning that the subset of $X_1\times X_2$ where there are not equal is a subset of a measurable set of measure zero). Is the following true :

  • For almost all $x_1\in X_1$, "$f(x_1,\cdot)=g(x_1,\cdot)$" almost everywhere in $X_2$.

In the answer is negative, does it become positive when we consider a completed product measure ? or when all measures are completed ?

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    $\begingroup$ Yes, by Fubini's theorem. $\endgroup$
    – Fan Zheng
    Commented Dec 3, 2016 at 3:59
  • $\begingroup$ Could you elaborate? Which version of Fubini theorem are you using to prove this? $\endgroup$
    – Jon-S
    Commented Dec 3, 2016 at 4:09
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    $\begingroup$ Just because you didn't get an answer on MSE doesn't make it suitable for MO. Look up Tonelli. $\endgroup$
    – Anthony Quas
    Commented Dec 3, 2016 at 4:17
  • $\begingroup$ I suppose you are the downvoter. But I did look up Tonelli. Could you please explain to me how does Tonelli trivially imply this ? $\endgroup$
    – Jon-S
    Commented Dec 3, 2016 at 4:19
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    $\begingroup$ Consider the characteristic function of the set $\{f\ne g\}$. $\endgroup$
    – Fan Zheng
    Commented Dec 3, 2016 at 5:01

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