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 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

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

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 limit -almost everywhere of a sequence of simple functions.

If is a continuous linear operator, and is Bochner-integrable, then is Bochner-integrable and integration and may be interchanged:

This also holds for closed operators, given that be itself integrable (which, via the criterion mentioned above is trivially true for bounded ).

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 and if

for almost every and then
as and
for all

If is Bochner integrable, then the inequality

holds for all In particular, the set function
defines a countably-additive -valued vector measure on which is absolutely continuous with respect to

Radon–Nikodym property

An important fact about the Bochner integral is that the Radon–Nikodym theorem fails to hold in general. This results in an important property of Banach spaces known as the Radon–Nikodym property. Specifically, if is a measure on then has the Radon–Nikodym property with respect to if, for every countably-additive vector measure on with values in which has bounded variation and is absolutely continuous with respect to there is a -integrable function such that

for every measurable set [1]

The Banach space has the Radon–Nikodym property if has the Radon–Nikodym property with respect to every finite measure. It is known that the space has the Radon–Nikodym property, but and the spaces for an open bounded subset of and for 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

References

  1. ^ Bárcenas, Diómedes (2003). "The Radon–Nikodym Theorem for Reflexive Banach Spaces" (PDF). Divulgaciones Matemáticas. 11 (1): 55–59 [pp. 55–56].