Bochner integral

The Bochner integral , named after Salomon Bochner , is a generalization of the Lebesgue integral to Banach space- valued functions.

definition

Let it be a -finite, complete measure space and a Banach space . ${\displaystyle (\Omega ,{\mathcal {A}},\mu )}$ ${\displaystyle \sigma }$ ${\displaystyle (B,\|\cdot \|)}$

The Bochner integral of a function is now defined as follows: ${\displaystyle \int _{\Omega }f\,{\rm {d}}\mu }$ ${\displaystyle f\colon \Omega \to B}$

We call functions of the shape as simple functions

${\displaystyle s(x)=\sum _{i=1}^{m}\alpha _{i}\chi _{X_{i}}(x)}$

with factors and measurable quantities , where their indicator function denotes. The integral of a simple function is now defined in an obvious way: ${\displaystyle \alpha _{i}\in B}$ ${\displaystyle X_{i}\in {\mathcal {A}}}$ ${\displaystyle \chi _{X_{i}}}$

${\displaystyle \int _{\Omega }s\,{\rm {d}}\mu :=\sum _{i=1}^{m}\alpha _{i}\mu (X_{i})}$,

where this is well-defined, i.e. independent of the concrete decomposition of . ${\displaystyle s}$

A function is -measurable , if a series is simple functions so that for -almost all true. ${\displaystyle f\colon \Omega \rightarrow B}$ ${\displaystyle \mu }$ ${\displaystyle (s_{n})_{n\in \mathbb {N} }}$ ${\displaystyle \lim _{n\to \infty }s_{n}(x)=f(x)}$ ${\displaystyle \mu }$ ${\displaystyle x\in \Omega }$

A measurable function is called Bochner integrable , if there is a sequence of simple functions such that ${\displaystyle \mu }$ ${\displaystyle f\colon \Omega \rightarrow B}$ ${\displaystyle (s_{n})_{n\in \mathbb {N} }}$

• ${\displaystyle \lim _{n\to \infty }s_{n}(x)=f(x)}$applies to almost everyone and ${\displaystyle \mu }$ ${\displaystyle x\in \Omega }$
• to each one exists with ${\displaystyle \varepsilon >0}$ ${\displaystyle n_{0}=n_{0}(\varepsilon )\in \mathbb {N} }$

${\displaystyle \int _{\Omega }\|s_{n}-s_{k}\|{\rm {d}}\mu <\varepsilon }$for everyone . ${\displaystyle n,k\geq n_{0}}$

In this case it is

${\displaystyle \int _{\Omega }f\,{\rm {d}}\mu :=\lim _{n\to \infty }\int _{\Omega }s_{n}\,{\rm {d}}\mu }$

well-defined, that is, independent of the choice of the concrete sequence with the above properties. If and , you write ${\displaystyle (s_{n})_{n\in \mathbb {N} }}$ ${\displaystyle M\in {\mathcal {A}}}$ ${\displaystyle f\colon M\rightarrow B}$

${\displaystyle \int _{M}f{\rm {d}}\mu :=\int _{\Omega }{\tilde {f}}{\rm {d}}\mu }$ With ${\displaystyle {\tilde {f}}(x):=\left\{{\begin{array}{ll}f(x)\ ,&{\rm {falls}}\ x\in M\ ,\\0\ ,&{\rm {falls}}\ x\in \Omega \setminus M,\\\end{array}}\right.}$

provided that Bochner can be integrated. ${\displaystyle {\tilde {f}}}$

Pettis' measurability theorem

The following sentence, which goes back to Billy James Pettis , characterizes the measurability: ${\displaystyle \mu }$

The function can only be measured if the following two conditions are met: ${\displaystyle f\colon \Omega \to B}$ ${\displaystyle \mu }$

• For every continuous linear functional , -is measurable. ${\displaystyle \phi \in B'}$ ${\displaystyle \phi \circ f\colon \Omega \to {\mathbb {K} }}$ ${\displaystyle \mu }$
• There is a - zero set , so that it is separable with regard to the standard topology . ${\displaystyle \mu }$ ${\displaystyle N\subset \Omega }$ ${\displaystyle f(\Omega \setminus N)\subset B}$

If there is a separable Banach space, then the second condition is automatically fulfilled and therefore unnecessary. Overall, the measurability of -valent functions with this theorem is reduced to the measurability of scalar functions. ${\displaystyle B}$ ${\displaystyle \mu }$ ${\displaystyle B}$ ${\displaystyle \mu }$

Bochner integrability

The following equivalent characterization of Bochner-integrable functions found by Bochner allows some classical results of Lebesgue's integration theory such as B. to transfer the theorem of the majorized convergence to the Bochner integral:

A measurable function is Bochner integrable if and only if Lebesgue can be integrated. ${\displaystyle \mu }$ ${\displaystyle f\colon \Omega \to B}$ ${\displaystyle \|f\|:\Omega \to \mathbb {R} }$

properties

This section is a Banach space and functions are integrable. ${\displaystyle B}$ ${\displaystyle f,g\colon \Omega \rightarrow B}$

Linearity

The Bochner integral is linear , that is, for Bochner integrable functions and any can also be integrated, and the following applies: ${\displaystyle f,g\colon \Omega \rightarrow B}$ ${\displaystyle \alpha ,\beta \in \mathbb {K} }$ ${\displaystyle \ \alpha f+\beta g\ }$

${\displaystyle \int _{\Omega }\alpha f+\beta g\,\mathrm {d} \mu =\alpha \cdot \int _{\Omega }f\,\mathrm {d} \mu +\beta \cdot \int _{\Omega }g\,\mathrm {d} \mu }$.

Concatenation with a continuous operator

Let it be a Banach space and a continuous linear operator . Then there is an integrable function and it applies ${\displaystyle D}$ ${\displaystyle T\in L(B,D)}$ ${\displaystyle Tf\colon \Omega \to D}$

${\displaystyle T\left(\int _{\Omega }f(x)\mathrm {d} \mu \right)=\int _{\Omega }T(f)(x)\mathrm {d} \mu }$.

Radon – Nikodym property

→ Main article : Radon-Nikodym property

The Radon-Nikodým theorem does not generally apply to the Bochner integral. Banach spaces, for which this theorem applies, are called Banach spaces with the Radon-Nikodym property . Reflexive rooms always have the Radon-Nikodym property.

Bochner Lebesgue rooms

If a -finite, complete measure space and a Banach space, then the space of the Bochner integrable functions is called a Bochner Lebesgue space, whereby, as usual, almost identical functions are identified. You get with the norm ${\displaystyle (\Omega ,{\mathcal {A}},\mu )}$ ${\displaystyle \sigma }$ ${\displaystyle (B,\|\cdot \|)}$ ${\displaystyle L^{1}(\Omega ,{\mathcal {A}},\mu ,B)}$ ${\displaystyle \Omega \rightarrow B}$ ${\displaystyle \mu }$

${\displaystyle \|f\|_{1}:=\int _{\Omega }\|f(\omega )\|\mathrm {d} \mu (\omega )}$

a Banach room. This can be described as the tensor product as follows . One calculates that through

${\displaystyle L^{1}(\Omega ,{\mathcal {A}},\mu )\times B\rightarrow L^{1}(\Omega ,{\mathcal {A}},\mu ,B),\,(f,\alpha )\mapsto f(\cdot )\alpha }$

a bilinear map is given that has an isometric isomorphism

${\displaystyle L^{1}(\Omega ,{\mathcal {A}},\mu ){\hat {\otimes }}_{\pi }B\cong L^{1}(\Omega ,{\mathcal {A}},\mu ,B)}$

where denotes the projective tensor product . ${\displaystyle {\hat {\otimes }}_{\pi }}$

See also

• Birkhoff integral
• Pettis integral

literature

• Herbert Amann, Joachim Escher : Analysis. Volume 3. Birkhäuser, Basel et al. 2001, ISBN 3-7643-6613-3 .
• Malempati M. Rao: Measure Theory and Integration (= Pure and Applied Mathematics. A Program of Monographs, Textbooks, and Lecture Notes. Vol. 265). 2nd edition, revised and expanded. Dekker, New York NY et al. 2004, ISBN 0-8247-5401-8 , p. 505 ff.

Web links

• Salomon Bochner : Integration of functions whose values ​​are the elements of a vector space. (PDF; 799 kB). In: Fundamenta Mathematicae . Vol. 20, 1933, pp. 262-276.
• VI Sobolev: Bochner integral. In: Encyclopaedia of Mathematics (English).
• Integral vector valued functions. In: Matroids Math Planet .

Individual evidence

1. ^ Herbert Amann, Joachim Escher: Analysis. Volume 3. 2001, note X.2.1 (a).
2. ^ Herbert Amann, Joachim Escher: Analysis. Volume 3. 2001, p. 65.
3. ^ Herbert Amann, Joachim Escher: Analysis. Volume 3. 2001, p. 87.
4. ^ Herbert Amann, Joachim Escher: Analysis. Volume 3. 2001, Corollary X.2.7.
5. ^ Herbert Amann, Joachim Escher: Analysis. Volume 3. 2001, p. 94.
6. ^ Herbert Amann, Joachim Escher: Analysis. Volume 3. 2001, p. 92.
7. ↑ Joseph Diestel, John Jerry Uhl: Vector Measures (= Mathematical Surveys. Vol. 15). American Mathematical Society, Providence RI 1977, ISBN 0-8218-1515-6 , Corollary III.2.13.
8. ^ Raymond A. Ryan: Introduction to Tensor Products of Banach Spaces , Springer-Verlag 2002, ISBN 1-85233-437-1 , example 2.19