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 $(X,\mathrm{\Sigma },\mu )$ 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 $$s(x)=\sum _{i=1}^n{\chi }_{E_i}(x)b_i,$$ where the $E_i$ are disjoint members of the $\sigma$-algebra $\mathrm{\Sigma },$ 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 $b_i\ne 0,$ then the simple function is integrable, and the integral is then defined by $$\underset{X}{\int }[\sum _{i=1}^n{\chi }_{E_i}(x)b_i]d\mu =\sum _{i=1}^n\mu (E_i)b_i$$ 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 $$\underset{n\to \mathrm{\infty }}{\lim }\underset{X}{\int }\Vert f-s_n{\Vert }_{B}d\mu =0,$$ where the integral on the left-hand side is an ordinary Lebesgue integral. In this case, the Bochner integral is defined by $$\underset{X}{\int }fd\mu =\underset{n\to \mathrm{\infty }}{\lim }\underset{X}{\int }{s}_{n}d\mu .$$ It can be shown that the sequence ${\left\{\underset{X}{\int }{s}_{n}d\mu \right\}}_{n=1}^{\mathrm{\infty }}$ 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 $\{s_n{\}}_{n=1}^{\mathrm{\infty }}.$ 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 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 $(X,\mathrm{\Sigma },\mu )$ is a measure space, then a Bochner-measurable function $f:X\to B$ is Bochner integrable if and only if $$\underset{X}{\int }\Vert f{\Vert }_B\mathrm{d}\mu <\mathrm{\infty }.$$ Here, a function $f:X\to B$ 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 $\mathrm{\Sigma }$. Equivalently, $f$ is the limit $\mu$-almost everywhere of a sequence of countably-valued simple functions.

Linear operators

If $T:B\to B^{\prime }$ is a continuous linear operator between Banach spaces $B$ and $B^{\prime }$, and $f:X\to B$ is Bochner integrable, then it is relatively straightforward to show that $Tf:X\to B^{\prime }$ is Bochner integrable and integration and the application of $T$ may be interchanged: $$\underset{E}{\int }Tf\mathrm{d}\mu =T\underset{E}{\int }f\mathrm{d}\mu$$ for all measurable subsets $E\in \mathrm{\Sigma }$. A non-trivially stronger form of this result, known as Hille's theorem, also holds for closed operators. If $T:B\to B^{\prime }$ is a closed linear operator between Banach spaces $B$ and $B^{\prime }$ and both $f:X\to B$ and $Tf:X\to B^{\prime }$ are Bochner integrable, then $$\underset{E}{\int }Tf\mathrm{d}\mu =T\underset{E}{\int }f\mathrm{d}\mu$$ for all measurable subsets $E\in \mathrm{\Sigma }$.

Dominated convergence theorem

A version of the dominated convergence theorem also holds for the Bochner integral. Specifically, if $f_n:X\to B$ is a sequence of measurable functions on a complete measure space tending almost everywhere to a limit function $f$, and if $$\Vert f_n(x){\Vert }_B\le g(x)$$ for almost every $x\in X$, and $g\in L^1(\mu )$, then $$\underset{E}{\int }\Vert f-f_n{\Vert }_B\mathrm{d}\mu \to 0$$ as $n\to \mathrm{\infty }$ and $$\underset{E}{\int }{f}_n\mathrm{d}\mu \to \underset{E}{\int }f\mathrm{d}\mu$$ for all $E\in \mathrm{\Sigma }$. If $f$ is Bochner integrable, then the inequality $${\Vert \underset{E}{\int }f\mathrm{d}\mu \Vert }_B\le \underset{E}{\int }\Vert f{\Vert }_B\mathrm{d}\mu$$ holds for all $E\in \mathrm{\Sigma }.$ In particular, the set function $$E\mapsto \underset{E}{\int }f\mathrm{d}\mu$$ 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 $(X,\mathrm{\Sigma }),$ then $B$ has the Radon–Nikodym property with respect to $\mu$ if, for every countably-additive vector measure $\gamma$ on $(X,\mathrm{\Sigma })$ 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 $$\gamma (E)=\underset{E}{\int }gd\mu$$ for every measurable set $E\in \mathrm{\Sigma }.$ 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 $L^{\mathrm{\infty }}(\mathrm{\Omega }),$ $L^1(\mathrm{\Omega }),$ for $\mathrm{\Omega }$ 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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• 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
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• van Dulst, D. (2001) [1994], "Vector measures", Encyclopedia of Mathematics, EMS Press