Fubini's theorem
Conditions for switching order of integration in calculus / From Wikipedia, the free encyclopedia
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In mathematical analysis, the Fubini 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:
In the previous formula, we use Lebesgue integrals. In general, the formula is not true if Riemann integrals are used, but it is true if the function is assumed to be continuous on the rectangle, and sometimes this weaker result is known as Fubini's theorem in multivariable calculus.
The Tonelli theorem, introduced by Leonida Tonelli in 1909 is similar but is applied to a non-negative measurable function rather than to an integrable function over its domain. The Fubini and Tonelli's theorems usually are combined and form the Fubini-Tonelli theorem, which gives the conditions under which it is possible to switch the order of integration in an iterated integral.
A related theorem is often called Fubini's theorem for infinite series,[1] which states that: if is a double-indexed sequence of real numbers, and if
is absolutely convergent, then
Although the Fubini theorem for infinite series is a special case of the more general Fubini's theorem, it is not appropriate to characterize the former as a logical consequence of the later. This is because some properties of measures, in particular subadditivity, are often proved using the Fubini theorem for infinite series.[2] In this case, the Fubini theorem for integrals is a logical consequence of the Fubini's theorem for infinite series.