Hypotheses of Tonelli's and Fubini's theorems

Let \(f\) be a Lebesgue measurable function on \(\RR^n\times\RR^k\).

• Tonelli. If \(f\) takes values in \([0,\infty]\), then \(\int_{\RR^{n+k}}f\) equals both iterated integrals of \(f\), as elements of \([0,\infty]\) (FT-4JRQX).

• Fubini. If \(f\in L^1(\RR^{n+k})\), then \(\int_{\RR^{n+k}}f\) equals both iterated integrals of \(f\), whose inner integrals exist for almost every value of the outer variable (T-4GPEF).