Kurzweil-Henstock Integral in Riesz spaces - Bentham Science

Kurzweil-Henstock Integral in Riesz Spaces, 25-41 25
2 Elementary Theory of Riesz Spaces
Abstract
In this chapter we deal with the fundamental properties of lattice groups and Riesz spaces. We introduce the concepts of
order and (D)-convergence, weak -distributivity and Egorov property and prove some related results. We deal also
with order bounded and order continuous linear functionals in the setting of Riesz spaces. Finally we introduce the
Maeda-Ogasawara-Vulikh representation theorem.
2.1 Lattice Ordered Groups
Throughout this chapter, given a< b
R , we will denote by [ ab , ] the closed interval
{ x R : a xb} and by ( ab , ) the open interval { x R : a< x< b}.
Definition 2.1 A partially ordered set R is a nonempty set endowed with a reflexive, transitive
and antisymmetric relation, denoted by .
Given a nonempty subset A R and an element s R,
we say that s is the supremum of A if for
every element a A we have a s,
and moreover we get s c whenever c R and c b for all
b A.
Analogously, given j R,
we say that j is the infimum of A if for each a A we have
a j,
and for all d R,
such that d b b A,
we get j d . In this case, we write s = sup A
and j = inf A respectively.
If is any nonempty set and ( x) is a family of elements in R , we denote also by x and
x , or sup x
and inf x,
the quantities sup{ x : } and inf { x : }
respectively, provided that they exist in R .
A partially ordered set R is said to be a lattice if for every two elements a , b R there exist in R
the supremum s:= a b ( = sup {a, b} ) and the infimum j:= a b ( = inf{ ab , }) . In a partially
ordered set R , we say that a nonempty subset A R is bounded from above, if there exists x A
such that a x,
a A;
bounded from below, if there exists y A such that a y,
a A;
bounded, if it is bounded both from above and from below.
A lattice R is said to be Dedekind complete if every nonempty subset of R , bounded from above
(with respect to the relation ), admits supremum in R , and every nonempty subset of R , bounded
from below, admits infimum in R . A Dedekind complete lattice R is said to be super Dedekind
complete if, for any nonempty set A R,
bounded from above, there exists a countable subset
A A,
such that sup A= sup A , and for every nonempty set B R,
bounded from below, there
exists a countable subset B B,
such that inf B= inf B .
An element a of a lattice R is said to be positive if a 0 . Two positive elements ab , R are
said to be disjoint or orthogonal if a b=
0 . A nonempty set A R is called a disjoint system if
every element a A is positive and a b=
0 ab , A.
Given any two elements a , b R,
we
say that a< b or b> a if a b and a b.
A unit of R is an element a such that a > 0 .
A lattice R is said to be laterally complete if every disjoint system A R has a supremum in
R . We say that R is universally complete if it is both Dedekind and laterally complete.
Definition 2.2 Let R be a lattice. A nonempty set C R is said to be directed upwards
[downwards ] if for every pair ab , C there exists c C such that c a,
c b [c a,
c b]
.
Antonio Boccuto / Beloslav Riean / Marta Vrábelová
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