Fubini's theorem

The Fubini's theorem is a set of integral calculus . It indicates under which conditions and how one can calculate multi-dimensional integrals with the help of one-dimensional integrals. This theorem was first proven in 1907 by Guido Fubini (1879–1943).

description

With the help of the Riemann integral or the Lebesgue integral one can define the integration of functions over multidimensional areas. The problem here is that these integrals are defined over a limit value by breaking down the area into small parts. However, this does not provide a useful, constructive method to calculate such integrals. In the case of one-dimensional integrals, this limit value formation can be avoided if an antiderivative can be found for the function to be integrated ( main theorem of differential and integral calculus ).

With the help of Fubini's theorem, multidimensional integrals can now be reduced to one-dimensional ones, which in turn can be calculated with the help of an antiderivative (if known). The theorem also states that the order of the one-dimensional integrations does not matter. This trick was naively used (before an exact definition of the integration calculation) as early as the 16th century and is known in the special case of volume calculations based on the Cavalieri principle .

Fubini's theorem for the Riemann integral

Be steady. ${\displaystyle f\colon [a,b]\times [c,d]\to \mathbb {R} }$

Then with is continuous and it applies ${\displaystyle F\colon [c,d]\to \mathbb {R} }$ ${\displaystyle F(y):=\int _{a}^{b}f(x,y)\,\mathrm {d} x}$

${\displaystyle \int _{c}^{d}F(y)\,\mathrm {d} y=\int _{c}^{d}\int _{a}^{b}f(x,y)\,\mathrm {d} x\,\mathrm {d} y=\int _{a}^{b}\int _{c}^{d}f(x,y)\,\mathrm {d} y\,\mathrm {d} x=\int \limits _{[a,b]\times [c,d]}f(x,y)\,\mathrm {d} (x,y)}$.

Fubini's theorem for the Lebesgue integral

Let us be and two- finite dimensional spaces and a measurable function that can be integrated with regard to the product dimensions , that is, it applies ${\displaystyle (\Omega _{1},{\mathcal {A}}_{1},\mu _{1})}$ ${\displaystyle (\Omega _{2},{\mathcal {A}}_{2},\mu _{2})}$ ${\displaystyle \sigma }$ ${\displaystyle f\colon \Omega _{1}\times \Omega _{2}\to \mathbb {R} }$ ${\displaystyle \mathrm {\mu } _{1}\otimes \mu _{2}}$

${\displaystyle \;\int \limits _{\Omega _{1}\times \Omega _{2}}|f|\,\mathrm {d} (\mu _{1}\otimes \mu _{2})<\infty \ }$

or it applies almost everywhere . ${\displaystyle f\geq 0}$

Then the function is for almost everyone ${\displaystyle y}$

${\displaystyle x\mapsto f(x,y)}$

and the function for almost everyone ${\displaystyle x}$

${\displaystyle y\mapsto f(x,y)}$

integrable or non-negative. You can therefore use the functions defined or defined by integration ${\displaystyle y}$ ${\displaystyle x}$

${\displaystyle x\mapsto \int \limits _{\Omega _{2}}f(x,y)~\mathrm {d} \mu _{2}(y)}$

${\displaystyle y\mapsto \int \limits _{\Omega _{1}}f(x,y)~\mathrm {d} \mu _{1}(x)}$

consider. These can also be integrated or are non-negative and it applies

${\displaystyle \;\int \limits _{\Omega _{1}\times \Omega _{2}}f~\mathrm {d} (\mu _{1}\otimes \mu _{2})=\int \limits _{\Omega _{2}}^{}\int \limits _{\Omega _{1}}^{}f(x,y)~\mathrm {d} \mu _{1}(x)~\mathrm {d} \mu _{2}(y)=\int \limits _{\Omega _{1}}^{}\int \limits _{\Omega _{2}}^{}f(x,y)~\mathrm {d} \mu _{2}(y)~\mathrm {d} \mu _{1}(x).}$

Tonelli's theorem (also Fubini-Tonelli's theorem)

A useful variation on this last movement is Tonelli's theorem . The ability to be integrated with regard to the product size is not a prerequisite here. It is enough that for the iterated integrals exist: ${\displaystyle |f|}$

Let be a real measurable function like above. If one of the two iterated integrals ${\displaystyle f(x,y)}$

${\displaystyle \;\int \limits _{\Omega _{2}}^{}\int \limits _{\Omega _{1}}^{}|f(x,y)|~\mathrm {d} \mu _{1}(x)~\mathrm {d} \mu _{2}(y)}$,

${\displaystyle \;\int \limits _{\Omega _{1}}^{}\int \limits _{\Omega _{2}}^{}|f(x,y)|~\mathrm {d} \mu _{2}(y)~\mathrm {d} \mu _{1}(x)}$

exists, then the other also exists, can be integrated with regard to the product size and the following applies: ${\displaystyle f(x,y)}$

${\displaystyle \;\int \limits _{\Omega _{1}\times \Omega _{2}}f~\mathrm {d} (\mu _{1}\otimes \mu _{2})=\int \limits _{\Omega _{2}}^{}\int \limits _{\Omega _{1}}^{}f(x,y)~\mathrm {d} \mu _{1}(x)~\mathrm {d} \mu _{2}(y)=\int \limits _{\Omega _{1}}^{}\int \limits _{\Omega _{2}}^{}f(x,y)~\mathrm {d} \mu _{2}(y)~\mathrm {d} \mu _{1}(x).}$

Inferences

By considering components by component, it immediately follows that Fubini's theorem applies not only to real-valued functions, but also to functions with values ​​in finite-dimensional real vector spaces. Since the field of complex numbers is a two-dimensional vector space, Fubini's theorem also applies to complex-valued functions or functions with values ​​in finite-dimensional vector spaces. ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$

Stochastics

With the help of Fubini's theorem, one can prove the following identities, which are used, for example, in stochastics.

• Let Lebesgue be integrable, then: ${\displaystyle f:\ [a,b]\times [a,b]\longrightarrow \mathbb {R} }$

${\displaystyle \int \limits _{a}^{b}{\Biggl (}\int \limits _{a}^{y}f(x,y)~\mathrm {d} x~{\Biggr )}\mathrm {d} y=\int \limits _{a}^{b}{\Biggl (}\int \limits _{x}^{b}f(x,y)~\mathrm {d} y~{\Biggr )}\mathrm {d} x.}$

• Let Lebesgue be integrable, then inductively follows: ${\displaystyle f:\ \mathbb {R} \longrightarrow \mathbb {R} }$

${\displaystyle \int \limits _{0}^{x}{\Biggl (}\int \limits _{0}^{x_{1}}\dots \int \limits _{0}^{x_{n-2}}{\Biggl (}\int \limits _{0}^{x_{n-1}}f(x_{n})\mathrm {d} x_{n}{\Biggr )}\mathrm {d} x_{n-1}\dots \mathrm {d} x_{2}{\Biggr )}\mathrm {d} x_{1}={\frac {1}{(n-1)!}}\int \limits _{0}^{x}(x-t)^{n-1}f(t)\mathrm {d} t.}$

Convolution of two functions

In addition, the kit provides a simple proof of the well-definedness the convolution of two functions: Be out of the -space . denote the Lebesgue measure . Define the function ${\displaystyle f,g}$ ${\displaystyle {\mathcal {L}}^{1}(\mathbb {R} ^{n},\mathbb {C} )}$ ${\displaystyle \lambda }$

${\displaystyle F\colon (\mathbb {R} ^{n})^{2}\to \mathbb {C} }$, . ${\displaystyle (x,y)\mapsto f(y)g(x-y)}$

Then applies

${\displaystyle \int _{\mathbb {R} ^{n}}\int _{\mathbb {R} ^{n}}|F(x,y)|\,\mathrm {d} \lambda (x)\,\mathrm {d} \lambda (y)\leq \int _{\mathbb {R} ^{n}}\left(\int _{\mathbb {R} ^{n}}|g(x-y)|\,\mathrm {d} \lambda (x)\right)|f(y)|\,\mathrm {d} \lambda (y)=\left(\int _{\mathbb {R} ^{n}}|f|\,\mathrm {d} \lambda \right)\cdot \left(\int _{\mathbb {R} ^{n}}|g|\,\mathrm {d} \lambda \right)=\lVert f\rVert _{1}\cdot \lVert g\rVert _{1}\in \mathbb {R} }$.

So, according to Fubini-Tonelli, the integral also exists

${\displaystyle \int _{\mathbb {R} ^{n}}\int _{\mathbb {R} ^{n}}|f(y)g(x-y)|\,\mathrm {d} \lambda (y)\,\mathrm {d} \lambda (x)}$

and is equal to the above integral.

In particular, the (measurable) functions , for almost every absolutely integrable. So the convolution of the functions and is given by ${\displaystyle F_{x}\colon \mathbb {R} ^{n}\to \mathbb {C} }$ ${\displaystyle y\mapsto F(x,y)}$ ${\displaystyle x\in \mathbb {R} ^{n}}$ ${\displaystyle f}$ ${\displaystyle g}$

${\displaystyle (f*g)(x):=\int _{\mathbb {R} ^{n}}F_{x}(y)\,\mathrm {d} \lambda (y)}$,

well defined.

In addition, the function is also included in, and it applies . ${\displaystyle f*g}$ ${\displaystyle {\mathcal {L}}^{1}(\mathbb {R} ^{n},\mathbb {C} )}$ ${\displaystyle \lVert f*g\rVert _{1}\leq \lVert f\rVert _{1}\cdot \lVert g\rVert _{1}}$

literature

• Jürgen Elstrodt : Measure and integration theory. 7th edition. Springer, Berlin / Heidelberg 2011, ISBN 978-3-642-17904-4 , Chapter V.
• Achim Klenke: Probability Theory . Springer-Verlag, Berlin Heidelberg 2008, ISBN 978-3-540-76317-8 , p. 279.
• Konrad Königsberger: Analysis 2 . 5th edition, Springer, Berlin 2004.

Individual evidence

1. ^ Fubini, Guido (1907), "Sugli integrali multipli", Rome. Acc. L. Rend. (5), 16 (1): 608-614.