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 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 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 is finite whenever b_i \neq 0, then the simple function is integrable, and the integral is then defined by 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 where the integral on the left-hand side is an ordinary Lebesgue integral. In this case, the Bochner integral is defined by It can be shown that the sequence 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 is a measure space, then a Bochner-measurable function is Bochner integrable if and only if Here, a function  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 is a continuous linear operator between Banach spaces B and B', and is Bochner integrable, then it is relatively straightforward to show that is Bochner integrable and integration and the application of T may be interchanged: for all measurable subsets. A non-trivially stronger form of this result, known as Hille's theorem, also holds for closed operators. If is a closed linear operator between Banach spaces B and B' and both and are Bochner integrable, then for all measurable subsets.

Dominated convergence theorem

A version of the dominated convergence theorem also holds for the Bochner integral. Specifically, if is a sequence of measurable functions on a complete measure space tending almost everywhere to a limit function f, and if for almost every x \in X, and, then as and for all. If f is Bochner integrable, then the inequality holds for all In particular, the set function 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 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 for every measurable set 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: It is known that the space \ell 1 has the Radon–Nikodym property, but c 0 and the spaces 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.