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:
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
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
If is bounded, the condition
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Bochner integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bochner_integral&oldid=11334