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Difference between revisions of "Fubini theorem"

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A theorem that establishes a connection between a multiple integral and a repeated one. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041870/f0418701.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041870/f0418702.png" /> are measure spaces with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041870/f0418703.png" />-finite complete measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041870/f0418704.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041870/f0418705.png" /> defined on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041870/f0418706.png" />-algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041870/f0418707.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041870/f0418708.png" />, respectively. If the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041870/f0418709.png" /> is integrable on the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041870/f04187010.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041870/f04187011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041870/f04187012.png" /> with respect to the product measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041870/f04187013.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041870/f04187014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041870/f04187015.png" />, then for almost-all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041870/f04187016.png" /> the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041870/f04187017.png" /> of the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041870/f04187018.png" /> is integrable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041870/f04187019.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041870/f04187020.png" />, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041870/f04187021.png" /> is integrable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041870/f04187022.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041870/f04187023.png" />, and one has the equality
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A theorem that establishes a connection between a multiple integral and a repeated one. Suppose that $(X,\mathfrak S_X,\mu_x)$ and $(Y,\mathfrak S_Y,\mu_y)$ are measure spaces with $\sigma$-finite complete measures $\mu_x$ and $\mu_y$ defined on the $\sigma$-algebras $\mathfrak S_X$ and $\mathfrak S_Y$, respectively. If the function $f(x,y)$ is integrable on the product $X\times Y$ of $X$ and $Y$ with respect to the product measure $\mu=\mu_x\times\mu_y$ of $\mu_x$ and $\mu_y$, then for almost-all $y\in Y$ the function $f(x,y)$ of the variable $x$ is integrable on $X$ with respect to $\mu_x$, the function $g(y)=\int_Xf(x,y)d\mu_x$ is integrable on $Y$ with respect to $\mu_y$, and one has the equality
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041870/f04187024.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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$$\int\limits_{X\times Y}f(x,y)d\mu=\int\limits_Yd\mu_y\int\limits_Xf(x,y)d\mu_x.\label{1}\tag{1}$$
  
Fubini's theorem is valid, in particular, for the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041870/f04187025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041870/f04187026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041870/f04187027.png" /> are the Lebesgue measures in the Euclidean spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041870/f04187028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041870/f04187029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041870/f04187030.png" /> respectively (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041870/f04187031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041870/f04187032.png" /> are natural numbers), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041870/f04187033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041870/f04187034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041870/f04187035.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041870/f04187036.png" /> is a Lebesgue-measurable function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041870/f04187037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041870/f04187038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041870/f04187039.png" />. Under these assumptions, formula (1) has the form
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Fubini's theorem is valid, in particular, for the case when $\mu_x$, $\mu_y$ and $\mu$ are the Lebesgue measures in the Euclidean spaces $\mathbf R^m$, $\mathbf R^n$ and $\mathbf R^{m+n}$ respectively ($m$ and $n$ are natural numbers), $X=\mathbf R^m$, $Y=\mathbf R^n$, $X\times Y=\mathbf R^m\times\mathbf R^n=\mathbf R^{m+n}$, and $f=f(x,y)$ is a Lebesgue-measurable function on $\mathbf R^{m+n}$, $x\in\mathbf R^m$, $y\in\mathbf R^n$. Under these assumptions, formula \eqref{1} has the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041870/f04187040.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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$$\iint\limits_{\mathbf R^{m+n}}f(x,y)d(x,y)=\int\limits_{\mathbf R^n}dy\int\limits_{\mathbf R^m}f(x,y)dx.\label{2}\tag{2}$$
  
In the case of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041870/f04187041.png" /> defined on an arbitrary Lebesgue-measurable set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041870/f04187042.png" />, in order to express the multiple integral in terms of a repeated one, one must extend <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041870/f04187043.png" /> by zero to the whole of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041870/f04187044.png" /> and apply (2). See also [[Repeated integral|Repeated integral]].
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In the case of a function $f$ defined on an arbitrary Lebesgue-measurable set $E\subset\mathbf R^{m+n}$, in order to express the multiple integral in terms of a repeated one, one must extend $f$ by zero to the whole of $\mathbf R^{m+n}$ and apply \eqref{2}. See also [[Repeated integral|Repeated integral]].
  
 
The theorem was established by G. Fubini [[#References|[1]]].
 
The theorem was established by G. Fubini [[#References|[1]]].

Latest revision as of 17:07, 14 February 2020

A theorem that establishes a connection between a multiple integral and a repeated one. Suppose that $(X,\mathfrak S_X,\mu_x)$ and $(Y,\mathfrak S_Y,\mu_y)$ are measure spaces with $\sigma$-finite complete measures $\mu_x$ and $\mu_y$ defined on the $\sigma$-algebras $\mathfrak S_X$ and $\mathfrak S_Y$, respectively. If the function $f(x,y)$ is integrable on the product $X\times Y$ of $X$ and $Y$ with respect to the product measure $\mu=\mu_x\times\mu_y$ of $\mu_x$ and $\mu_y$, then for almost-all $y\in Y$ the function $f(x,y)$ of the variable $x$ is integrable on $X$ with respect to $\mu_x$, the function $g(y)=\int_Xf(x,y)d\mu_x$ is integrable on $Y$ with respect to $\mu_y$, and one has the equality

$$\int\limits_{X\times Y}f(x,y)d\mu=\int\limits_Yd\mu_y\int\limits_Xf(x,y)d\mu_x.\label{1}\tag{1}$$

Fubini's theorem is valid, in particular, for the case when $\mu_x$, $\mu_y$ and $\mu$ are the Lebesgue measures in the Euclidean spaces $\mathbf R^m$, $\mathbf R^n$ and $\mathbf R^{m+n}$ respectively ($m$ and $n$ are natural numbers), $X=\mathbf R^m$, $Y=\mathbf R^n$, $X\times Y=\mathbf R^m\times\mathbf R^n=\mathbf R^{m+n}$, and $f=f(x,y)$ is a Lebesgue-measurable function on $\mathbf R^{m+n}$, $x\in\mathbf R^m$, $y\in\mathbf R^n$. Under these assumptions, formula \eqref{1} has the form

$$\iint\limits_{\mathbf R^{m+n}}f(x,y)d(x,y)=\int\limits_{\mathbf R^n}dy\int\limits_{\mathbf R^m}f(x,y)dx.\label{2}\tag{2}$$

In the case of a function $f$ defined on an arbitrary Lebesgue-measurable set $E\subset\mathbf R^{m+n}$, in order to express the multiple integral in terms of a repeated one, one must extend $f$ by zero to the whole of $\mathbf R^{m+n}$ and apply \eqref{2}. See also Repeated integral.

The theorem was established by G. Fubini [1].

References

[1] G. Fubini, "Sugli integrali multipli" , Opere scelte , 2 , Cremonese (1958) pp. 243–249 Zbl 38.0343.02
How to Cite This Entry:
Fubini theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fubini_theorem&oldid=28194
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article