Measurable subsets of a product have measurable almost-everywhere slices, whose measures integrate to the total measure.Suggest a slogan
Let \(n=n_1+n_2\), let \(m\) denote Lebesgue measure on each \(\RR^k\), and let \(E\subseteq\RR^{n}=\RR^{n_1}\times\RR^{n_2}\) be Lebesgue measurable. For \(x\in\RR^{n_1}\), let \(E_x \coloneqq \theset{y \in \RR^{n_2} \suchthat (x,y) \in E}\). Then:
For almost every \(x\in \RR^{n_1}\), the slice \(E_x\) is measurable in \(\RR^{n_2}\).
The function \[ F\colon \RR^{n_1} \to [0,\infty], \qquad x \mapsto m(E_x) = \int_{\RR^{n_2}} \chi_{E_x}(y) \,dy, \] defined for almost every \(x\), is measurable, and \[ m(E) = \int_{\RR^{n_1}} m(E_x) \,dx = \int_{\RR^{n_1}} \int_{\RR^{n_2}} \chi_{E_x}(y) \,dy \,dx . \]