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In mathematics, the Bochner integral, named for Salomon Bochner, extends the definition of a multidimensional Lebesgue integral to functions that take values in a Banach space, as the limit of integrals of simple functions.

Definition

Let $(X,\Sigma ,\mu )$ be a measure space, and $B$ be a Banach space, and define a measurable function $f:X\to B$ . When $B=\mathbb {R}$ , we have the standard Lebesgue integral $\int _{X}fd\mu$ , and when $B=\mathbb {R} ^{n}$ , we have the standard multidimensional Lebesgue integral $\int _{X}{\vec {f}}d\mu$ . For generic Banach spaces, the Bochner integral extends the above cases.

First, define a simple function to be any finite sum of the form $s(x)=\sum _{i=1}^{n}\chi _{E_{i}}(x)b_{i},$ where the $E_{i}$ are disjoint members of the $\sigma$ -algebra $\Sigma ,$ the $b_{i}$ are distinct elements of $B,$ and χ_E is the characteristic function of $E.$ If $\mu \left(E_{i}\right)$ is finite whenever $b_{i}\neq 0,$ then the simple function is integrable, and the integral is then defined by $\int _{X}\left[\sum _{i=1}^{n}\chi _{E_{i}}(x)b_{i}\right]\,d\mu =\sum _{i=1}^{n}\mu (E_{i})b_{i}$ exactly as it is for the ordinary Lebesgue integral.

A measurable function $f:X\to B$ is Bochner integrable if there exists a sequence of integrable simple functions $s_{n}$ such that $\lim _{n\to \infty }\int _{X}\|f-s_{n}\|_{B}\,d\mu =0,$ where the integral on the left-hand side is an ordinary Lebesgue integral.

In this case, the Bochner integral is defined by $\int _{X}f\,d\mu =\lim _{n\to \infty }\int _{X}s_{n}\,d\mu .$

It can be shown that the sequence $\left\{\int _{X}s_{n}\,d\mu \right\}_{n=1}^{\infty }$ is a Cauchy sequence in the Banach space $B,$ hence the limit on the right exists; furthermore, the limit is independent of the approximating sequence of simple functions $\{s_{n}\}_{n=1}^{\infty }.$ These remarks show that the integral is well-defined (i.e independent of any choices). It can be shown that a function is Bochner integrable if and only if it lies in the Bochner space $L^{1}.$

Properties

Elementary properties

Many of the familiar properties of the Lebesgue integral continue to hold for the Bochner integral. Particularly useful is Bochner's criterion for integrability, which states that if $(X,\Sigma ,\mu )$ is a measure space, then a Bochner-measurable function $f\colon X\to B$ is Bochner integrable if and only if $\int _{X}\|f\|_{B}\,\mathrm {d} \mu <\infty .$

Here, a function $f\colon X\to B$ is called Bochner measurable if it is equal $\mu$ -almost everywhere to a function $g$ taking values in a separable subspace $B_{0}$ of $B$ , and such that the inverse image $g^{-1}(U)$ of every open set $U$ in $B$ belongs to $\Sigma$ . Equivalently, $f$ is the limit $\mu$ -almost everywhere of a sequence of countably-valued simple functions.

Linear operators

If $T\colon B\to B'$ is a continuous linear operator between Banach spaces $B$ and $B'$ , and $f\colon X\to B$ is Bochner integrable, then it is relatively straightforward to show that $Tf\colon X\to B'$ is Bochner integrable and integration and the application of $T$ may be interchanged: $\int _{E}Tf\,\mathrm {d} \mu =T\int _{E}f\,\mathrm {d} \mu$ for all measurable subsets $E\in \Sigma$ .

A non-trivially stronger form of this result, known as Hille's theorem, also holds for closed operators.^[1] If $T\colon B\to B'$ is a closed linear operator between Banach spaces $B$ and $B'$ and both $f\colon X\to B$ and $Tf\colon X\to B'$ are Bochner integrable, then $\int _{E}Tf\,\mathrm {d} \mu =T\int _{E}f\,\mathrm {d} \mu$ for all measurable subsets $E\in \Sigma$ .

Dominated convergence theorem

A version of the dominated convergence theorem also holds for the Bochner integral. Specifically, if $f_{n}\colon X\to B$ is a sequence of measurable functions on a complete measure space tending almost everywhere to a limit function $f$ , and if $\|f_{n}(x)\|_{B}\leq g(x)$ for almost every $x\in X$ , and $g\in L^{1}(\mu )$ , then $\int _{E}\|f-f_{n}\|_{B}\,\mathrm {d} \mu \to 0$ as $n\to \infty$ and $\int _{E}f_{n}\,\mathrm {d} \mu \to \int _{E}f\,\mathrm {d} \mu$ for all $E\in \Sigma$ .

If $f$ is Bochner integrable, then the inequality $\left\|\int _{E}f\,\mathrm {d} \mu \right\|_{B}\leq \int _{E}\|f\|_{B}\,\mathrm {d} \mu$ holds for all $E\in \Sigma .$ In particular, the set function $E\mapsto \int _{E}f\,\mathrm {d} \mu$ defines a countably-additive $B$ -valued vector measure on $X$ which is absolutely continuous with respect to $\mu$ .

Radon–Nikodym property

An important fact about the Bochner integral is that the Radon–Nikodym theorem fails to hold in general, and instead is a property (the Radon–Nikodym property) defining an important class of ″nice″ Banach spaces.

Specifically, if $\mu$ is a measure on $(X,\Sigma ),$ then $B$ has the Radon–Nikodym property with respect to $\mu$ if, for every countably-additive vector measure $\gamma$ on $(X,\Sigma )$ with values in $B$ which has bounded variation and is absolutely continuous with respect to $\mu ,$ there is a $\mu$ -integrable function $g:X\to B$ such that $\gamma (E)=\int _{E}g\,d\mu$ for every measurable set $E\in \Sigma .$ ^[2]

The Banach space $B$ has the Radon–Nikodym property if $B$ has the Radon–Nikodym property with respect to every finite measure.^[2] Equivalent formulations include:

Bounded discrete-time martingales in $B$ converge a.s.^[3]
Functions of bounded-variation into $B$ are differentiable a.e.^[4]
For every bounded $D\subseteq B$ , there exists $f\in B^{*}$ and $\delta \in \mathbb {R} ^{+}$ such that $\{x:f(x)+\delta >\sup {f(D)}\}\subseteq D$ has arbitrarily small diameter.^[3]

It is known that the space $\ell _{1}$ has the Radon–Nikodym property, but $c_{0}$ and the spaces $L^{\infty }(\Omega ),$ $L^{1}(\Omega ),$ for $\Omega$ an open bounded subset of $\mathbb {R} ^{n},$ and $C(K),$ for $K$ an infinite compact space, do not.^[5] Spaces with Radon–Nikodym property include separable dual spaces (this is the Dunford–Pettis theorem)^{[citation needed]} and reflexive spaces, which include, in particular, Hilbert spaces.^[2]

Probability

The Bochner integral is used in probability theory for handling random variables and stochastic processes that take values in a Banach space. The classical convergence theorems—such as the dominated convergence theorem—when applied to the Bochner integral, generalizes laws of large numbers and central limit theorems for sequences of Banach-space valued random variables. Such integrals are central to the analysis of distributions in functional spaces and have applications in fields such as stochastic calculus, statistical field theory ,and quantum field theory.

Let $(\Omega ,{\mathcal {F}},\mathbb {P} )$ be a probability space, and consider a random variable $X\colon \Omega \to B$ taking values in a Banach space $B$ . When $X$ is Bochner integrable, its expectation is defined by $E[X]=\int _{\Omega }X\,d\mathbb {P} ,$ which inherits the usual linearity and continuity properties of the classical expectation.

Stochastic process

Consider $\{X_{t}\}_{t\in T}$ , a stochastic process that is Banach-space valued. The Bochner integral allows us to define the mean function $\mu (t)=E[X_{t}]=\int _{\Omega }X_{t}\,d\mathbb {P} ,$ whenever each $X_{t}$ is Bochner integrable. This approach is useful in stochastic partial differential equations, where each $X_{t}$ is a random element in a functional space.

In martingale theory, a sequence $\{M_{n}\}_{n\geq 1}$ of $B$ -valued random variables is called a martingale with respect to a filtration $\{{\mathcal {F}}_{n}\}_{n\geq 1}$ if each $M_{n}$ is ${\mathcal {F}}_{n}$ -measurable, Bochner integrable, and satisfies $E[M_{n+1}\mid {\mathcal {F}}_{n}]=M_{n}.$ The Bochner integral ensures that conditional expectations are well-defined in this Banach space setting.

Gaussian measure

The Bochner integral allows the definition of Gaussian measures on a Banach space, where one often encounters integrals of the form $\int _{B}\langle x,b^{*}\rangle \,d\mu (x),$ where $b^{*}\in B^{*}$ and $\langle \cdot ,\cdot \rangle$ denotes the dual pairing.

References

^ Diestel, Joseph; Uhl, Jr., John Jerry (1977). Vector Measures. Mathematical Surveys. Vol. 15. American Mathematical Society. doi:10.1090/surv/015. ISBN 978-0-8218-1515-1. (See Theorem II.2.6)
^ ^a ^b ^c Bárcenas, Diómedes (2003). "The Radon–Nikodym Theorem for Reflexive Banach Spaces" (PDF). Divulgaciones Matemáticas. 11 (1): 55–59 [pp. 55–56].
^ ^a ^b Bourgin 1983, pp. 31, 33. Thm. 2.3.6-7, conditions (1,4,10).
^ Bourgin 1983, p. 16. "Early workers in this field were concerned with the Banach space property that each $X$ -valued function of bounded variation on $[0,1]$ be differentiable almost surely. It turns out that this property (known as the Gelfand-Fréchet property) is also equivalent to the RNP [Radon-Nikodym Property]."
^ Bourgin 1983, p. 14.

Bochner, Salomon (1933), "Integration von Funktionen, deren Werte die Elemente eines Vektorraumes sind" (PDF), Fundamenta Mathematicae, 20: 262–276, doi:10.4064/fm-20-1-262-176
Bourgin, Richard D. (1983). Geometric Aspects of Convex Sets with the Radon-Nikodým Property. Lecture Notes in Mathematics 993. Vol. 993. Berlin: Springer-Verlag. doi:10.1007/BFb0069321. ISBN 3-540-12296-6.
Cohn, Donald (2013), Measure Theory, Birkhäuser Advanced Texts Basler Lehrbücher, Springer, doi:10.1007/978-1-4614-6956-8, ISBN 978-1-4614-6955-1
Yosida, Kôsaku (1980), Functional Analysis, Classics in Mathematics, vol. 123, Springer, doi:10.1007/978-3-642-61859-8, ISBN 978-3-540-58654-8
Diestel, Joseph (1984), Sequences and Series in Banach Spaces, Graduate Texts in Mathematics, vol. 92, Springer, doi:10.1007/978-1-4612-5200-9, ISBN 978-0-387-90859-5
Diestel; Uhl (1977), Vector measures, American Mathematical Society, ISBN 978-0-8218-1515-1
Hille, Einar; Phillips, Ralph (1957), Functional Analysis and Semi-Groups, American Mathematical Society, ISBN 978-0-8218-1031-6
Lang, Serge (1993), Real and Functional Analysis (3rd ed.), Springer, ISBN 978-0387940014
Sobolev, V. I. (2001) [1994], "Bochner integral", Encyclopedia of Mathematics, EMS Press
van Dulst, D. (2001) [1994], "Vector measures", Encyclopedia of Mathematics, EMS Press

[1] Diestel, Joseph; Uhl, Jr., John Jerry (1977). Vector Measures. Mathematical Surveys. Vol. 15. American Mathematical Society. doi:10.1090/surv/015. ISBN 978-0-8218-1515-1. (See Theorem II.2.6)

[Reflex-2] Bárcenas, Diómedes (2003). "The Radon–Nikodym Theorem for Reflexive Banach Spaces" (PDF). Divulgaciones Matemáticas. 11 (1): 55–59 [pp. 55–56].

[Equiv-3] Bourgin 1983, pp. 31, 33. Thm. 2.3.6-7, conditions (1,4,10).

[4] Bourgin 1983, p. 16. "Early workers in this field were concerned with the Banach space property that each $X$ -valued function of bounded variation on $[0,1]$ be differentiable almost surely. It turns out that this property (known as the Gelfand-Fréchet property) is also equivalent to the RNP [Radon-Nikodym Property]."

[5] Bourgin 1983, p. 14.

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Integrals
Types of integrals	Riemann integral Lebesgue integral Burkill integral Bochner integral Daniell integral Darboux integral Henstock–Kurzweil integral Haar integral Hellinger integral Khinchin integral Kolmogorov integral Lebesgue–Stieltjes integral Pettis integral Pfeffer integral Riemann–Stieltjes integral Regulated integral
Integration techniques	Substitution Trigonometric Euler Weierstrass By parts Partial fractions Euler's formula Inverse functions Changing order Reduction formulas Parametric derivatives Differentiation under the integral sign Laplace transform Contour integration Laplace's method Numerical integration Simpson's rule Trapezoidal rule Risch algorithm
Improper integrals	Gaussian integral Dirichlet integral Fermi–Dirac integral complete incomplete Bose–Einstein integral Frullani integral Common integrals in quantum field theory
Stochastic integrals	Itô integral Russo–Vallois integral Stratonovich integral Skorokhod integral
Miscellaneous	Basel problem Euler–Maclaurin formula Gabriel's horn Integration Bee Proof that 22/7 exceeds π Volumes Washers Shells

Analysis in topological vector spaces
Basic concepts	Abstract Wiener space Classical Wiener space Bochner space Convex series Cylinder set measure Infinite-dimensional vector function Matrix calculus Vector calculus
Derivatives	Differentiable vector–valued functions from Euclidean space Differentiation in Fréchet spaces Fréchet derivative Total Functional derivative Gateaux derivative Directional Generalizations of the derivative Hadamard derivative Holomorphic Quasi-derivative
Measurability	Besov measure Cylinder set measure Canonical Gaussian Classical Wiener measure Measure like set functions infinite-dimensional Gaussian measure Projection-valued Vector Bochner / Weakly / Strongly measurable function Radonifying function
Integrals	Bochner Direct integral Dunford Gelfand–Pettis/Weak Regulated Paley–Wiener
Results	Cameron–Martin theorem Inverse function theorem Nash–Moser theorem Feldman–Hájek theorem No infinite-dimensional Lebesgue measure Sazonov's theorem Structure theorem for Gaussian measures
Related	Crinkled arc Covariance operator
Functional calculus	Borel functional calculus Continuous functional calculus Holomorphic functional calculus
Applications	Banach manifold (bundle) Convenient vector space Choquet theory Fréchet manifold Hilbert manifold

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