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 $$s(x) = \sum_{i=1}^n \chi_{E_i}(x) b_i,$$ where the $$E_i$$ are disjoint members of the $$\sigma$$-algebra $$\Sigma,$$ the $$b_i$$ are distinct elements of $$B,$$ and χE 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 $$ \left\{\int_Xs_n\,d\mu \right\}_{n=1}^{\infty} $$ 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 $$L^1.$$

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 $$T f \colon X \to B'$$ is Bochner integrable and integration and the application of $$T$$ may be interchanged: $$\int_E T f \, \mathrm{d} \mu = T \int_E f \, \mathrm{d} \mu$$ 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 $$T f \colon X \to B'$$ are Bochner integrable, then $$\int_E T f \, \mathrm{d} \mu = T \int_E f \, \mathrm{d} \mu$$ 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 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 $$\R^n,$$ 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.