Bochner-Integral

The Bochner integral, named after Salomon Bochner, is a generalization of the Lebesgue integral of Banach space - valued functions.

Definition

There are a - finite, complete measure space and a Banach space.

The Bochner integral of a function is now defined as follows:

As a simple function we denote functions of the form

With factors and measurable quantities, with their indicator function called. The integral of a simple function is now defined in an obvious way:

And this is well-defined, ie is independent of the specific decomposition of.

A function is called - measurable if there exists a sequence of simple functions such that all true of almost -.

A - measurable function is called Bochner integrable if there exists a sequence of simple functions such that

• For - almost all valid and
• To each one exists with

In this case,

Well-defined, ie independent of the choice of the specific sequence with the above properties. If and, one writes

Is provided Bochner - integrable.

Messbarkeitssatz of Pettis

The following goes back to Billy James Pettis theorem characterizes the measurability:

The function is exactly then - measurable if the following two conditions are met:

• For every continuous linear functional is - measurable.
• There is a -null set, so that is separable with respect to the norm topology.

If a separable Banach space, then the second condition is automatically satisfied and thus expendable. Overall, the measurability - valued functions with this set is attributed to the measurability of scalar functions.

Bochner integrability

The following discovered by Bochner integrable functions equivalent characterization Bochner allows some classical results of Lebesgue integration theory, for example, to transfer as the set of the dominated convergence on the Bochner integral:

A - measurable function is exactly then Bochner - integrable if

Properties

This section is a Banach space and are integrable functions.

Linearity

Bochner the integral is linear, that is, for Bochner - integrable functions, and any is also integrated, and the following applies:

Concatenation with a continuous operator

It is a Banach space and a continuous linear operator. Then is an integrable function and it is

Radon - Nikodym property

The Radon - Nikodym theorem does not apply to the Bochner integral in general. Banach spaces for which this rate applies are called Banach spaces with the Radon - Nikodym property. Reflexive spaces always have the Radon - Nikodym property.