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# Fubini's theorem

Fubini's theorem states that if
$\int\int \left|f(x,y)\right|\,d(x,y)<\infty$
the integral being taken with respect to a product measure on the space over which the pair of variables (x, y) ranges, then
$\int\left(\int f(x,y)\,dy\right)\,dx=\int\left(\int f(x,y)\,dx\right)\,dy=\int\int f(x,y)\,d(x,y)$
the first two integrals being iterated integrals, and the third being an integral with respect to a product measure.

A standard example showing that the assumption of finiteness cannot be dispensed with is

$\int_0^1\int_0^1\frac{x^2-y^2}{(x^2+y^2)^2}\,dx\,dy.$
Obviously the sign gets reversed if the order of iterated integration gets reversed, i.e., if "dy dx" replaces "dx dy". But the value of the integral is not zero, and so the values of the two iterated integrals differ from each other. Therefore, by the contrapositive of Fubini's theorem, we must have
$\int_0^1\int_0^1\frac{\left|x^2-y^2\right|}{(x^2+y^2)^2}\,dx\,dy=\infty.$

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