Let \(f\colon\RR^{n}\times \RR^k\to[0,\infty]\) be Lebesgue measurable on \(\RR^{n+k}\). For \(y\in\RR^k\), let \(f^y\colon\RR^n\to[0,\infty]\) be the slice \(f^y(x) \coloneqq f(x, y)\). Then:
For almost every \(y\in \RR^k\), the slice \(f^y\) is measurable on \(\RR^n\).
The function \(F(y) \coloneqq \int_{\RR^n} f^y(x) \, dx\), defined for almost every \(y\), is measurable on \(\RR^k\).
In \([0,\infty]\), \[ \int_{\RR^{n+k}} f(x,y) \, d(x,y) = \int_{\RR^k} \qty{\int_{\RR^n} f(x,y) \,dx}\, dy . \]
The same statements hold with the roles of \(x\) and \(y\) exchanged, so the two iterated integrals are equal.
The hypotheses are that \(f\) is nonnegative and measurable; \(f\) need not be integrable, and all three integrals in (3) may be \(+\infty\). Fubini’s theorem instead assumes \(f\in L^1(\RR^{n+k})\) and allows \(f\) to take values of both signs.