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Inmathematics, theBochner integral, named forSalomon Bochner, extends the definition of a multidimensionalLebesgue integral to functions that take values in aBanach space, as the limit of integrals ofsimple functions.

The Bochner integral provides the mathematical foundation for extensions of basicintegral transforms into more abstract spaces, vector-valued functions, and operator spaces. Examples of such extensions include vector-valuedLaplace transforms and abstractFourier transforms.^[1]

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

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

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

In this case, theBochner integral is defined by$${\displaystyle {\int }_{X}fd\mu =\underset{n\to \mathrm{\infty }}{lim}{\int }_X{s}_{n}d\mu .}$$

It can be shown that the sequence${\displaystyle {\left\{{\int }_X{s}_{n}d\mu \right\}}_{n=1}^{\mathrm{\infty }}}$ is aCauchy sequence in the Banach space${\displaystyle B,}$ hence the limit on the right exists; furthermore, the limit is independent of the approximating sequence of simple functions${\displaystyle \{s_n{\}}_{n=1}^{\mathrm{\infty }}.}$ 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 theBochner space${\displaystyle L^1.}$

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${\displaystyle (X,\mathrm{\Sigma },\mu )}$ is a measure space, then a Bochner-measurable function${\displaystyle f:X\to B}$ is Bochner integrable if and only if

Here, a function${\displaystyle f:X\to B}$ is calledBochner measurable if it is equal${\displaystyle \mu }$-almost everywhere to a function${\displaystyle g}$ taking values in a separable subspace${\displaystyle B_0}$ of${\displaystyle B}$, and such that the inverse image${\displaystyle g^{-1}(U)}$ of every open set${\displaystyle U}$ in${\displaystyle B}$ belongs to${\displaystyle \mathrm{\Sigma }}$. Equivalently,${\displaystyle f}$ is the limit${\displaystyle \mu }$-almost everywhere of a sequence of countably-valued simple functions.

If${\displaystyle T:B\to B^{\prime }}$ is a continuous linear operator between Banach spaces${\displaystyle B}$ and${\displaystyle B^{\prime }}$, and${\displaystyle f:X\to B}$ is Bochner integrable, then it is relatively straightforward to show that${\displaystyle Tf:X\to B^{\prime }}$ is Bochner integrable and integration and the application of${\displaystyle T}$ may be interchanged:$${\displaystyle {\int }_{E}Tf\mathrm{d}\mu =T{\int }_{E}f\mathrm{d}\mu }$$for all measurable subsets${\displaystyle E\in \mathrm{\Sigma }}$.

A non-trivially stronger form of this result, known asHille's theorem, also holds forclosed operators.^[2] If${\displaystyle T:B\to B^{\prime }}$ is a closed linear operator between Banach spaces${\displaystyle B}$ and${\displaystyle B^{\prime }}$ and both${\displaystyle f:X\to B}$ and${\displaystyle Tf:X\to B^{\prime }}$ are Bochner integrable, then$${\displaystyle {\int }_{E}Tf\mathrm{d}\mu =T{\int }_{E}f\mathrm{d}\mu }$$for all measurable subsets${\displaystyle E\in \mathrm{\Sigma }}$.

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

If${\displaystyle f}$ is Bochner integrable, then the inequality$${\displaystyle {\Vert {\int }_{E}f\mathrm{d}\mu \Vert }_B\le {\int }_E\Vert f{\Vert }_B\mathrm{d}\mu }$$holds for all${\displaystyle E\in \mathrm{\Sigma }.}$ In particular, the set function$${\displaystyle E\mapsto {\int }_{E}f\mathrm{d}\mu }$$defines a countably-additive${\displaystyle B}$-valuedvector measure on${\displaystyle X}$ which isabsolutely continuous with respect to${\displaystyle \mu }$.

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

Specifically, if${\displaystyle \mu }$ is a measure on${\displaystyle (X,\mathrm{\Sigma }),}$ then${\displaystyle B}$ has the Radon–Nikodym property with respect to${\displaystyle \mu }$ if, for every countably-additivevector measure${\displaystyle \gamma }$ on${\displaystyle (X,\mathrm{\Sigma })}$ with values in${\displaystyle B}$ which hasbounded variation and is absolutely continuous with respect to${\displaystyle \mu ,}$ there is a${\displaystyle \mu }$-integrable function${\displaystyle g:X\to B}$ such that$${\displaystyle \gamma (E)={\int }_{E}gd\mu }$$for every measurable set${\displaystyle E\in \mathrm{\Sigma }.}$^[3]

The Banach space${\displaystyle B}$ has the Radon–Nikodym property if${\displaystyle B}$ has the Radon–Nikodym property with respect to every finite measure.^[3] Equivalent formulations include:

• Bounded discrete-timemartingales in${\displaystyle B}$ converge a.s.^[4]
• Functions of bounded-variation into${\displaystyle B}$ are differentiable a.e.^[5]
• For every bounded${\displaystyle D\subseteq B}$, there exists${\displaystyle f\in B^{\ast }}$ and${\displaystyle \delta \in {\mathbb{R}}^{+}}$ such that$${\displaystyle \{x:f(x)+\delta >supf(D)\}\subseteq D}$$ has arbitrarily small diameter.^[4]

It is known that the space${\displaystyle {\ell }_1}$ has the Radon–Nikodym property, but${\displaystyle c_0}$ and the spaces${\displaystyle L^{\mathrm{\infty }}(\mathrm{\Omega }),}$${\displaystyle L^1(\mathrm{\Omega }),}$ for${\displaystyle \mathrm{\Omega }}$ an open bounded subset of${\displaystyle {\mathbb{R}}^n,}$ and${\displaystyle C(K),}$ for${\displaystyle K}$ an infinite compact space, do not.^[6] Spaces with Radon–Nikodym property include separable dual spaces (this is theDunford–Pettis theorem)^{[citation needed]} andreflexive spaces, which include, in particular,Hilbert spaces.^[3]

The Bochner integral is used inprobability 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, generalize 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 asstochastic calculus,statistical field theory, andquantum field theory.

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

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

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

The Bochner integral allows one to define moments for theGaussian measure on a Banach space. If the Bochner integral exists, then it is equivalent to thePettis integral${\displaystyle \eta }$ defined by$${\displaystyle ⟨\eta ,b^{\ast }⟩={\int }_B⟨x,b^{\ast }⟩d\mu (x),}$$where${\displaystyle b^{\ast }\in B^{\ast }}$ and${\displaystyle ⟨\cdot ,\cdot ⟩}$ denotes thedual pairing.

There are several extensions of the Bochner integral to functions${\displaystyle f:X\to E}$ with values in somelocally convex space${\displaystyle E}$ (1975 Rybakov,^[7] 1981 Blondia,^[8] 2015 Beckmann and Deitmar^[9]). The extension by Beckmann and Deitmar uses the original approach of Bochner but generalized tonets and they distinguish three cases of assumptions on the locally convex space:^[10]

• ${\displaystyle E}$ is complete,
• ${\displaystyle E}$ isquasi-complete space and the function${\displaystyle f:X\to E}$ is bounded,
• ${\displaystyle E}$ ist quasi-complete and the measure is finite.

Beckmann and Deitmar use the term ofBochner-approximability as a condition for defining the Bochner integral. A function is Bochner-approximable if there exists a net${\displaystyle (s_j{)}_{j\in J}}$ of simple functions such that for every continuous seminorm${\displaystyle p}$ on${\displaystyle E}$

${\displaystyle {\int }_{X}p(f-s_j)d\mu \to 0}$

They provide several equivalent characterizations of this property.

Blondia defines the extensions to locally convex spaces as follows^[11]

Let${\displaystyle (X,\mathrm{\Sigma },\mu )}$ be${\displaystyle \sigma }$-finite, complete measure space and${\displaystyle (E,\mathcal{P})}$ a complete Hausdorff locally convex space whose topology is induced by the family of seminorms${\displaystyle \mathcal{P}}$. A function${\displaystyle f:X\to E}$ is called Bochner integrable or strong integrable if there exists a sequence${\displaystyle (f_n)}$ such that

• ${\displaystyle f_n(\omega )\to f(\omega )}$ for${\displaystyle \omega \in X}$${\displaystyle \mu }$-almost surely
• ${\displaystyle p(f(\omega )-f_n(\omega ))\in L^1(X;\mathbb{R})}$ for each${\displaystyle n\in \mathbb{N}}$ and alle${\displaystyle p\in \mathcal{P}}$, i.e.

${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{\int }_{X}p(f(\omega )-f_n(\omega ))d\mu =0.}$

• ${\displaystyle {\int }_A{f}_n(\omega )d\mu }$ converges for each measurable subset${\displaystyle A}$ of${\displaystyle X}$.

• Bochner space – Type of topological space
• Bochner measurable function
• Pettis integral
• Vector measure – Generalization of finite measure to Banach spaces
• Weakly measurable function

• 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{{citation}}:ISBN / Date incompatibility (help)
• 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

• Definitions of mathematical integration
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