Bochner integral

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

An integral of a function with values in a Banach space with respect to a scalar measure. It belongs to the so-called strong integrals (cf. Strong integral).

Let be the vector space of functions , , with values in a Banach space , given on a space with a countably-additive scalar measure on a -algebra of subsets of . A function is called simple if A function is called strongly measurable if there exists a sequence of simple functions with almost-everywhere with respect to the measure on . In such a case the scalar function is -measurable. For the simple function  A function is said to be Bochner integrable if it is strongly measurable and if for some approximating sequence of simple functions The Bochner integral of such a function over a set is where is the characteristic function of , and the limit is understood in the sense of strong convergence in . This limit exists, and is independent of the choice of the approximation sequence of simple functions.

Criterion for Bochner integrability: For a strongly-measurable function to be Bochner integrable it is necessary and sufficient for the norm of this function to be integrable, i.e. The set of Bochner-integrable functions forms a vector subspace of , and the Bochner integral is a linear operator on this subspace.

Properties of Bochner integrals:

1) 2) A Bochner integral is a countably-additive -absolutely continuous set-function on the -algebra , i.e. if , and if , uniformly for .

3) If almost-everywhere with respect to the measure on , if almost-everywhere with respect to on , and if , then and 4) The space is complete with respect to the norm (cf. Convergence in norm) 5) If is a closed linear operator from a Banach space into a Banach space and if then If is bounded, the condition is automatically fulfilled, .

The Bochner integral was introduced by S. Bochner . Equivalent definitions were given by T. Hildebrandt  and N. Dunford (the -integral).

How to Cite This Entry:
Bochner integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bochner_integral&oldid=11334
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article