Fubini's theorem
Conditions for switching order of integration in calculus / From Wikipedia, the free encyclopedia
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In mathematical analysis, Fubini's theorem is a result that gives the conditions under which it is possible to compute a double integral by using an iterated integral, introduced by Guido Fubini in 1907. It states that if a function is (Lebesgue) integrable on a rectangle , then one can evaluate the double integral as an iterated integral:
Here all integrals are Lebesgue integrals. Fubini's theorem is not true as stated for the Riemann integral, but it is true if the function is assumed to be continuous on the rectangle, and sometimes this weaker result is called Fubini's theorem in multivariable calculus.
Tonelli's theorem, introduced by Leonida Tonelli in 1909, is similar, but applies to a non-negative measurable function rather than one integrable over their domains. Fubini's and Tonelli's theorems combine to the Fubini-Tonelli theorem (see below), which allows one to switch the order of integration in an iterated integral under certain conditions.
A related theorem is often called Fubini's theorem for infinite series,[1] which states that if is a doubly-indexed sequence of real numbers, and if is absolutely convergent, then
Although Fubini's theorem for infinite series is a special case of the more general Fubini's theorem, it is not appropriate to characterize it as a logical consequence of Fubini's theorem. This is because some properties of measures, in particular sub-additivity, are often proved using Fubini's theorem for infinite series.[2] In this case, Fubini's general theorem is a logical consequence of Fubini's theorem for infinite series.