Bochner integral

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, [3]–[5].

The Bochner integral was introduced by S. Bochner [1]. Equivalent definitions were given by T. Hildebrandt [2] and N. Dunford (the -integral).

References

[1]	S. Bochner, "Integration von Funktionen, deren Werte die Elemente eines Vektorraumes sind" Fund. Math. , 20 (1933) pp. 262–276
[2]	T.H. Hildebrandt, "Integration in abstract spaces" Bull. Amer. Math. Soc. , 59 (1953) pp. 111–139
[3]	K. Yosida, "Functional analysis" , Springer (1980) pp. Chapt. 8, §1
[4]	E. Hille, R.S. Phillips, "Functional analysis and semi-groups" , Amer. Math. Soc. (1957)
[5]	N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958)

Comments

A simple function is also called a step function. A good recent textbook on integrals with values in a Banach space is [a1]; [a4] is specifically about the Bochner integral.

References

[a1]	J. Diestel, J.J. Uhl jr., "Vector measures" , Math. Surveys , 15 , Amer. Math. Soc. (1977)
[a2]	A.C. Zaanen, "Integration" , North-Holland (1967)
[a3]	N. Bourbaki, "Elements of mathematics. Integration" , Addison-Wesley (1975) pp. Chapt.6;7;8 (Translated from French)
[a4]	J. Mikusiński, "The Bochner integral" , Acad. Press (1978)