Kurzweil-Henstock Integral in Riesz spaces - Bentham Science

8 Choquet and ipo Integrals
Kurzweil-Henstock Integral in Riesz Spaces, 111-133 111
Abstract: In this chapter we introduce some integrals for real-valued maps with respect to Riesz space-valued set
functions, which are not necessarily finitely additive, but in general can be simply only increasing. First we deal with
the ipo (symmetric) integral and prove the monotone and Lebesgue dominated convergence theorems, Fatou’s lemma
and the submodular theorem.
Moreover we introduce the Choquet (asymmetric) integral, giving in particular some applications to the weak and
strong laws of large numbers in the context of Riesz spaces.
8.1 Symmetric Integral
In this chapter we deal with some recent research and results about Choquet-type integrals for realvalued
(or extended real-valued) functions, with respect to set functions with values in a
Dedekind complete Riesz space , and which can be -additive, finitely additive or simply only
increasing. In the literature, in the case , there are substantially two kinds of integrals in this
context: the symmetric and the asymmetric integral (see also [73]). Here we investigate in detail
only the symmetric integral, because our main recent researches in this topic deal with the case in
which is finitely additive (here the two integrals will coincide) and, when is only monotone,
just (mainly) with the symmetric integral.
In [32] we introduced a "monotone-type" (that is, Choquet-type) integral for real-valued maps,
with respect to finitely additive positive set functions, with values in a Dedekind complete Riesz
space. In [98], a Choquet-type integral for real-valued functions with respect to Riesz-space-valued
"capacities" , that is monotone set functions (not necessarily finitely additive), is investigated.
We introduce a ipo-type integral, that is "symmetric" integral, for real-valued functions with
respect to Riesz-space-valued capacities, we investigate the fundamental properties and prove some
convergence theorems (for real-valued capacities see also [249] and [204], pp. 152-176).
Throughout this paragraph, we always suppose that is a Dedekind complete Riesz space. In
some suitable cases, we add to two extra elements, which we call and , extending
ordering and operations, having the same role as the usual and with the real numbers (see
also [15, 136]). We denote with the symbol the set .
We now extend to general directed nets the concept of -convergence.
A directed net is called -net if , that is if it is decreasing and . We
say that the directed net is -convergent to , if
and in this case we will write
Definition 8.1 Let be an arbitrary nonempty set, and The class
is called the upper set system of
Antonio Boccuto / Beloslav Riean / Marta Vrábelová
All rights reserved - © 2009 Bentham Science Publishers Ltd.