(after Fubini, Guido 1879-1943) Theorem of integral calculus which, in general, states that the order of integration of a multiple integral does not affect the integral's result. For example, in integrating a well-defined function f(x,y) over a rectangle defined by {a<=x<=b; c<=y<=d}, it doesn't matter whether the internal nested integral is taken over {a,b} with respect to x, or over {c,d} with respect to y. This technique often allows difficult integrals in certain forms to be transformed into forms which may be much easier (or simply possible) to evaluate.

Moreover, besides stating that integration for iterated integrals of functions that are continuously integrable over the region of integration may be evaluated in any order, Fubini states that the double or triple integral of any function over any region "nice enough" may be evaluated as the iterated integral. "Nice enough" meaning that the region can be broken into regions that can be described by two end values and bounded by continuous curves, over which the function is continously integrable. The function need not be continuously integrable over the region as a whole, it may be discontinuous along the graphs of finitely many functions.

Properly speaking, Fubini's theorem relates to the measure theoretic integral, not the Riemann integral commonly taught in undergraduate Mathematics curricula. The Riemann-integral equivalent is essentially that given above; however the conditions attached to changing the order of integration are truly horrific. As usual, the "real" integral does it in a much neater (and more intuitive, outrageously enough!) manner.

The theorem proper explains how (and when!) to convert integrals by some product measure to an iterated integral by each component in turn. I'll be using dx×dy as my product measure, with dx and dy as the components. This is more convenient for those accustomed to using Lebesgue measure, and will also appear familiar to people who know Riemann integrals. Note, however, that the theorem applies equally well to the product of any two measures.

Theorem. Let f(x,y) be an integrable function according to the measure dx×dy. Then for almost any value of y,
I(y) = ∫ f(x,y)dx
exists. Furthermore, I(y) is itself integrable and
I = ∫ f(x,y) dx×dy = ∫ I(y) dy = ∫ (∫ f(x,y) dx) dy.
By switching x and y and noting we still get I, we get the lax formulation:
Corollary. If f(x,y) is integrable then
∫ (∫ f(x,y) dx) dy = ∫ (∫ f(x,y) dy) dx.

Note the simple condition for switching order of integration: the function must be integrable as a function of 2 variables. This is (unfortunately for people who don't use it) only true for the Lebesgue integral.
