In mathematical analysis
, Fubini's theorem
, named after Guido Fubini
, states that if
the integral being taken with respect to a product measure on the space over , where A and B are complete measure spaces, then
the first two integrals being iterated integrals with respect to two measures respectively, and the third being an integral with respect to a product of these two measures. Also,
the third integral being with respect to a product measure.
If the above integral of the absolute value is not finite, then the two iterated integrals may actually have different values. See below for an illustration of this possibility.
Another version of Fubini's theorem states that if A and B are sigma-finite measure spaces, not necessarily complete, and if either
Thus we have
Fubini's theorem implies that since these two iterated integrals differ, the integral of the absolute value must be ∞.
then the two iterated integrals
may have different finite values.
Strong versions of Fubini's theorem
The existence of strengthenings of Fubini's theorem, where the function is no longer assumed to be measurable but merely that the two iterated integrals are well defined and exist, is independent of the standard Zermelo–Fraenkel axioms of set theory. Martin's axiom implies that there exists a function on the unit square whose iterated integrals are not equal, while a variant of Freiling's axiom of symmetry implies that in fact a strong Fubini-type theorem for [0, 1] does hold, and whenever the two iterated integrals exist they are equal. See List of statements undecidable in ZFC.