Let \(f\colon\RR^n\times\RR^m\to[0,\infty]\) be Lebesgue measurable on \(\RR^{n+m}\). For \(x\in\RR^n\) let \(f_x\colon\RR^m\to[0,\infty]\) be the slice \(f_x(y)\coloneqq f(x,y)\). Then:
for almost every \(x\in\RR^n\), the slice \(f_x\) is Lebesgue measurable on \(\RR^m\);
the function \(F(x)\coloneqq\int_{\RR^m} f(x,y)\,dy\in[0,\infty]\), defined for almost every \(x\), is Lebesgue measurable on \(\RR^n\);
\[ \int_{\RR^{n+m}} f = \int_{\RR^n} F(x)\,dx = \int_{\RR^n}\qty{\int_{\RR^m} f(x,y)\,dy}dx \in[0,\infty], \] and the same holds with the roles of \(x\) and \(y\) exchanged [@Fol13].
Moreover, if \(E\subseteq\RR^{n}\times\RR^m\) is Lebesgue measurable, then for almost every \(x\in\RR^n\) the slice \(E_x\coloneqq\theset{y\in\RR^m \suchthat (x, y) \in E}\) is Lebesgue measurable.