Editing Fubini's theorem (section)

{{short description|Conditions for switching order of integration in calculus}}
{{for|the Fubini theorem for [[Meagre set#Terminology|category]]|Kuratowski–Ulam theorem}}
In [[mathematical analysis]], '''Fubini's theorem''' characterizes the conditions under which it is possible to compute a [[double integral]] by using an [[iterated integral]]. It was introduced by [[Guido Fubini]] in 1907.  The theorem states that if a function is [[Lebesgue integral|Lebesgue integrable]] on a rectangle <math>X\times Y</math>, then one can evaluate the double integral as an iterated integral:<math display="block">\, \iint\limits_{X\times Y} f(x,y)\,\text{d}(x,y) = \int_X\left(\int_Y f(x,y)\,\text{d}y\right)\text{d}x=\int_Y\left(\int_X f(x,y) \, \text{d}x \right)  \text{d}y.</math> 
This formula is generally not true for the [[Riemann integral]], but it is true if the function is continuous on the rectangle. In [[multivariable calculus]], this weaker result is sometimes also called Fubini's theorem, although it was already known by [[Leonhard Euler]].

'''Tonelli's 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 theorems are usually combined and form the Fubini-Tonelli theorem, which gives the conditions under which it is possible to switch the [[Order of integration (calculus)|order of integration]] in an iterated integral.

A related theorem is often called '''Fubini's theorem for infinite series''',<ref>{{citation | first=Terence | last=Tao | title=Analysis I | pages=188 | year=2016 | publisher=Springer | isbn=9789811017896}}</ref> although it is due to [[Alfred Pringsheim]].<ref>{{cite book|author1=E T Whittaker|author2=G N Watson|title=A course of modern analysis|publisher=Cambridge University Press|year=1902}}</ref> It states that if <math display="inline">\{a_{m,n}\}_{m=1,n=1}^{\infty}</math> is a double-indexed sequence of real numbers, and if <math display="inline">\displaystyle \sum_{(m,n)\in \N\times \N} a_{m,n} </math> is absolutely convergent, then

: <math> \sum_{(m,n)\in\N\times \N}a_{m,n} = \sum_{m=1}^\infty\sum_{n=1}^{\infty} a_{m,n} = \sum_{n=1}^\infty \sum_{m=1}^\infty a_{m,n}. </math>

Although Fubini's theorem for infinite series is a special case of the more general Fubini's theorem, it is not necessarily appropriate to characterize the former as being proven by the latter because the properties of measures needed to prove Fubini's theorem proper, in particular subadditivity of measure, may be proven using Fubini's theorem for infinite series.<ref>{{citation|first=Halsey|last=Royden|title=Real Analysis|pages=34|year=2010|publisher=Prentice Hall |isbn=9780131437470}}</ref>