Kurzweil-Henstock Integral in Riesz spaces - Bentham Science

Kurzweil-Henstock Integral in Topological Spaces Kurzweil-Henstock Integral in Riesz Spaces 63
Example 5.7 The compact interval with the usual topology is a Hausdorff compact
topological space. The family of all compact subintervals (including singletons and ) is
separating, and defined by is additive and regular.
Lemma 5.8 If is separating, then to any gauge there exists a -fine partition.
Proof: Let us consider . We want to prove that the space is decomposable, i.e. that there
exists a partition of such that If not, then one of the
elements of (denote it by ) is not decomposable. Similarly there is an indecomposable
, etc. Evidently Let . Since is separating,
Since , there exists such that . But then the singleton is a
partition of . This is a contradiction with the assumption that is indecomposable.
Remark 5.9 If is a compact metric space, is a semiring of subsets of such that to any
and every neighborhood of there exists such that (where,
given any subset of any topological space, denotes its interior in the topological sense) and
is separating, then in correspondence with any set
also [217], Lemma 1.2., p. 154 and Proposition 1.7., p. 156).
there exists a -fine partition (see
Definition 5.10 (see also [217]) A function is -integrable (or, in short,
integrable) if there exists such that to any there exist a gauge such that
for every partition . Here the entity
is called Riemann sum or integral sum.
Evidently the number is determined uniquely. It will be denoted by
Proposition 5.11 The integral is a linear positive functional.
Proof: The linearity follows by the identity
The positivity follows by the implication
Proposition 5.12 (Bolzano-Cauchy condition) A function is integrable if and only if the
following condition holds:
To any there exists a gauge such that