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
$s(x) = \sum_^n \chi_(x) b_i$
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_^n \chi_(x) b_i\right, d\mu = \sum_^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
$\lim_\int_X \, f-s_n\, _B\,d\mu = 0,$
where the integral on the left-hand side is an ordinary Lebesgue integral.

In this case, the Bochner integral is defined by
$\int_X f\, d\mu = \lim_\int_X s_n\, d\mu.$

It can be shown that the sequence $\left\_^$ 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 $\_^.$ 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
$\int_X \, f\, _B\, \mathrm \mu < \infty.$

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 $g^(U)$ of every open set $U$ in $B$ belongs to $\Sigma$. Equivalently, $f$ is the limit $\mu$-almost everywhere of a sequence of 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 $T f \colon X \to B'$ is Bochner integrable and integration and the application of $T$ may be interchanged:
$\int_X T f \, \mathrm \mu = T \int_X f \, \mathrm \mu$
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 $T f \colon X \to B'$ are Bochner integrable, then
$\int_X T f \, \mathrm \mu = T \int_X f \, \mathrm \mu$
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
$\int_X \, f-f_n\, _B \, \mathrm \mu \to 0$
as $n \to \infty$ and
$\int_X f_n\, \mathrm \mu \to \int_X f \, \mathrm \mu$
for all $E \in \Sigma$.

If $f$ is Bochner integrable, then the inequality
$\left\, \int_E f \, \mathrm \mu\right\, _B \leq \int_E \, f\, _B \, \mathrm \mu$
holds for all $E \in \Sigma.$ In particular, the set function
$E\mapsto \int_E f\, \mathrm \mu$
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 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.. Thm. 2.3.6-7, conditions (1,4,10).
* Functions of bounded-variation into $B$ are differentiable a.e.
* For every bounded $D\subseteq B$, there exists $f\in B^*$ and $\delta\in\mathbb^+$ such that $\\subseteq D$ has arbitrarily small diameter.

It is known that the space $\ell_1$ has the Radon–Nikodym property, but $c_0$ and the spaces $L^(\Omega),$ $L^1(\Omega),$ for $\Omega$ an open bounded subset of $\R^n,$ 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

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References

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{{Analysis in topological vector spaces

Definitions of mathematical integration
Integral representations
Topological vector spaces