Fubini's theorem

In mathematical analysis, Fubini's theorem, named in honor of Guido Fubini, states that if

Related Topics:
Mathematical analysis - Guido Fubini

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:int_{A imes B} |f(x,y)|,d(x,y)

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

the integral being taken with respect to a product measure on the space over A imes B, then

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:int_Aleft(int_B f(x,y),dy ight),dx=int_Bleft(int_A f(x,y),dx ight),dy=int_{A imes B} f(x,y),d(x,y),

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

the first two integrals being iterated integrals, and the third being an integral with respect to a product measure. Also,

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:int_A f(x), dx int_B g(y), dy = int_{A imes B} f(x)g(y),d(x,y)

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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. For an example, see an elegant rearrangement of a conditionally convergent iterated integral.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~