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. Let be a measure space, and be a Banach space. The Bochner integral of a function 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 are disjoint members of the -algebra the are distinct elements of and χE is the characteristic function of If is finite whenever 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 is Bochner integrable if there exists a sequence of integrable simple functions 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 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 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 -almost everywhere to a function taking values in a separable subspace of , and such that the inverse image of every open set in belongs to . Equivalently, is the limit -almost everywhere of a sequence of countably-valued simple functions. If is a continuous linear operator between Banach spaces and , and is Bochner integrable, then it is relatively straightforward to show that is Bochner integrable and integration and the application of may be interchanged: for all measurable subsets .