Kurzweil-Henstock Integral in Riesz spaces - Bentham Science

6 Convergence Theorems
Kurzweil-Henstock Integral in Riesz Spaces, 71-83 71
Abstract: We continue to investigate the topics of Chapter 5, following the approach there introduced, and prove some
versions of the Henstock Lemma, the Beppo Levi and the Lebesgue dominated convergence theorem.
Note that the involved measures and the domain of our functions can be even unbounded.
6.1 Elementary Properties
Antonio Boccuto / Beloslav Riean / Marta Vrábelová
All rights reserved - © 2009 Bentham Science Publishers Ltd.
In the book we have exposed the elementary theory of the Kurzweil-Henstock integral (Chapter 1),
then basic facts about the Kurzweil-Henstock integral for functions from a compact interval to a
Riesz space (Chapter 3). Of course, in Chapter 5 there have been considered functions defined on a
compact topological space. In this chapter we deal with convergence theorems (monotone
convergence theorem and Lebesgue dominated convergence theorem) for the Kurzweil-Henstock
integral in an abstract setting, for functions defined in a compact topological space , satisfying
suitable properties, and with values in a Dedekind complete Riesz space, with respect to a positive
Riesz space-valued measure , which can assume even the value . A particular case is
an interval (possibly unbounded) of the extended real line, or the whole of , and
the Lebesgue measure (this case was investigated in [26]). We continue the investigation started in
[24], in which is -valued, and in which some other kinds of convergence theorems were
demonstrated. Our results given here are proved in [29] and extend the ones in [227], Chapter 5,
which were proved in the case in which is finite, and the ones of [170], which were proved in
the case , where is as above, and all the involved Riesz spaces coincide with the
(eventually extended) real line. A similar Kurzweil-Henstock type integral was investigated in [36]
for Banach space-valued maps.
From now on, in this chapter we shall always suppose that is a Dedekind complete weakly -
distributive Riesz space.
Assumption 6.1 Let , be as in Chapter 5, and let us consider a positive mapping
Definition 6.2 We say that is additive, if
whenever .
Let be the -algebra of all Borel subsets of . We say that a positive set function
is regular, if to any there exists a regulator such that for every there exist
compact, open and such that , .
The concepts of gauge, partition, decomposition, -fineness and separating family are given
analogously as in Chapter 5.