Kurzweil-Henstock Integral in Riesz spaces - Bentham Science

Integration in Metric Semigroups Kurzweil-Henstock Integral in Riesz Spaces 199
Definition 13.2 A fuzzy number or fuzzy set is a function satisfying the following
conditions:
(j) there exists such that ;
(jj) the -cut set is convex for ;
(jjj) is upper semi-continuous, i.e. any -cut is a closed subset of ;
(jv) the support of the function is a compact set.
Any real number can be identified with a fuzzy number in the following way:
i.e. , and , if .
The set of all fuzzy numbers is denoted by .
We now endow with a metric and a linear structure (see also [28, 282]). We define the
Hausdorff distance on the set of all compact possibly degenerate intervals in :
Let . It is easy to check that, for every , there exist , , , (depending
on ) such that , . So, for , set
Using this definition becomes a complete metric space.
To define a linear structure on , recall that every fuzzy number is completely determined by
its -cuts. Hence, for any , and , set
(here, ).
Finally, we note that is not a group, but only a semigroup (see also [28]), in fact let
be defined by the formula:
Then is given by