Let \(f\colon\RR^n\times\RR^k\to\CC\) be Lebesgue measurable on \(\RR^{n+k}\), and suppose that one of the iterated integrals of \(\abs{f}\) is finite: \[ \int_{\RR^k}\qty{\int_{\RR^n}\abs{f(x,y)}\,dx}dy<\infty \quad\text{or}\quad \int_{\RR^n}\qty{\int_{\RR^k}\abs{f(x,y)}\,dy}dx<\infty. \] Then \(f\in L^1(\RR^{n+k})\), and \[ \int_{\RR^{n+k}} f(x,y)\,d(x,y)=\int_{\RR^k}\qty{\int_{\RR^n} f(x,y)\,dx}dy=\int_{\RR^n}\qty{\int_{\RR^k} f(x,y)\,dy}dx, \] where the inner integrals exist for almost every value of the outer variable.
By Tonelli's theorem applied to the nonnegative measurable function \(\abs{f}\), \(\int_{\RR^{n+k}}\abs{f}\) equals each iterated integral of \(\abs{f}\), so it is finite and \(f\in L^1(\RR^{n+k})\). Fubini’s theorem for \(f\in L^1(\RR^{n+k})\) then gives the equality of \(\int_{\RR^{n+k}}f\) with both iterated integrals of \(f\).