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Let $A$ and $B$ be rectangles in $\mathbb{R}^k$ and $\mathbb{R}^n$ respectively. Let $S$ be a set contained in $A\times B$. For each $y\in B$, let $$S_y=\{x\mid x\in A\text{ and }(x,y)\in S\}.$$ We call $S_y$ a cross-section of $S$. Show that if $S$ is Jordan measurable, and if $S_y$ is Jordan measurable for each $y\in B$, then $$v(S)=\int_{y\in B}v(S_y).$$

This certainly reminds me of Fubini's theorem. I write the desired equation in the form $$\int_S1=\int_{y\in B}\int_{S_y}1$$

If $S$ were equal to the rectangle $A\times B$, $S_y$ would be equal to $A$ for all $y$, and this would be true by Fubini. But here $S$ is an arbitrary subset of $A\times B$, and instead of an arbitrary function $f$, we have here the constant function $f\equiv 1$. How can we proceed?

Mika H.
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  • In the more general setting of Lebesgue integration, this question asks about conditional probabilities and disintegration. Consider random variables $Z=(X,Y)$. If we show that a regular conditional probability $P(Z\in S|Y=y)$ can be chosen so that for a given $y$ it is concentrated on the event ${Y=y}$, i.e. $P(Z\in S|Y=y)=P(Y\in S_y|Y=y)$, then the requested result will follow. Some authors define the concept of disintegration, which is equivalent to the conditional probability plus the concentration requirement. In nice Borel spaces, like $\mathbb{R}^n$, this can be satisfied. – paperskilltrees Dec 22 '22 at 22:26
  • This question is duplicated and the answer is given here. Disregard my previous comment, which would make more sense if your inner integration were with respect to a $y$-dependent measure. – paperskilltrees Dec 22 '22 at 23:47

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