Bochner integral

In mathematics, the Bochner integral, named for Salomon Bochner, extends the definition of 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. The Bochner integral of a function $f:X\to B$ is defined in much the same way as the Lebesgue integral. First, define a simple function to be any finite sum of the form

$$s_{n}(x)=\sum _{i=1}^{n}\chi _{E_{i}}(x)b_{i},$$

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. 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 $E\in \Sigma .$

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

• Bounded discrete-time martingales in $B$ converge a.s.
• Functions of bounded-variation into $B$ are differentiable a.e.
• 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.

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. Spaces with Radon–Nikodym property include separable dual spaces (this is the Dunford–Pettis theorem) and reflexive spaces, which include, in particular, Hilbert spaces.

See also

• Bochner space – Type of topological space
• Bochner measurable function
• Pettis integral
• Vector measure
• Weakly measurable function

References

• Bochner, Salomon (1933), "Integration von Funktionen, deren Werte die Elemente eines Vektorraumes sind" (PDF), Fundamenta Mathematicae, 20: 262–276
• Bourgin, Richard D. (1983). Geometric Aspects of Convex Sets with the Radon-Nikodým Property. Lecture Notes in Mathematics 993. Berlin: Springer-Verlag. doi:10.1007/BFb0069321. ISBN 3-540-12296-6.
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• 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
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• 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