Fubini's theorem [SEP] named after
Fubini's theorem
In mathematical analysis Fubini's theorem, introduced by Guido Fubini in 1907, is a result that gives conditions under which it is possible to compute a double integral by using iterated integral. One may switch the order of integration if the double integral yields a finite answer when the integrand is replaced by its absolute value.
As a consequence it allows the order of integrations to be changed in iterated integral.
Fubini's theorem implies that two iterated integrals are equal to the corresponding double integral across its integrands
and a non-negative measurable function "f" for which the double integral of |"f"| is zero but the two iterated integrals have different values; see the section on counterexamples below for an example of this. Tonelli's theorem and the Fubini–Tonelli theorem (stated below) can fail on non σ-finite spaces even for the maximal product measure.
Tonelli's theorem for non-negative measurable function.
Tonelli's theorem (named after Leonida Tonelli) is a successor of Fubini's theorem. The conclusion of