Fubini theorem

A theorem that establishes a connection between a multiple integral and a repeated one. Suppose that and are measure spaces with -finite complete measures and defined on the -algebras and , respectively. If the function is integrable on the product of and with respect to the product measure of and , then for almost-all the function of the variable is integrable on with respect to , the function is integrable on with respect to , and one has the equality

(1)

Fubini's theorem is valid, in particular, for the case when , and are the Lebesgue measures in the Euclidean spaces , and respectively ( and are natural numbers), , , , and is a Lebesgue-measurable function on , , . Under these assumptions, formula (1) has the form

(2)

In the case of a function defined on an arbitrary Lebesgue-measurable set , in order to express the multiple integral in terms of a repeated one, one must extend by zero to the whole of and apply (2). See also Repeated integral.

The theorem was established by G. Fubini [1].

References

[1]	G. Fubini, "Sugli integrali multipli" , Opere scelte , 2 , Cremonese (1958) pp. 243–249