Kurzweil-Henstock Integral in Riesz spaces - Bentham Science

(SL)-Integral Kurzweil-Henstock Integral in Riesz Spaces 135
Moreover, if is a mapping, and is a partition or decomposition as above, we
denote by the quantity
Definition 9.4 A function
Lebesgue measure zero.
is called gage if the set has
We now endow the set of all gages with the following ordering. Given two gages and , we say
that if and only if for all .
Definition 9.5 If is a gage, we say that a decomposition is -fine if
and for all .
Remark 9.6 We note that there exist some gages , without -fine partitions. However for all
gages there exists a -fine decomposition (for a proof, see [212], Teorema 1, p. 259).
Definition 9.7 Let , and . We say that a function is of class
or has property on if for every set of Lebesgue measure zero and
there exists a map such that for any -fine decomposition
of , with and , we have:
We say that is of class if it is of class on .
The following properties hold:
Proposition 9.8
1) The functions satisfying property form a linear space.
2) Property implies continuity.
3) If , , are such that is of class on each , then is of class also on
.
Definition 9.9 Let be the set of all gages, defined on . We say that is
-integrable on if there exists a function of class such that, for every
, there exists such that
whenever is a -fine decomposition of .
The function will be called weak primitive of .
In this case we put (by definition ) .
The integral is well-defined: indeed we have the following:
(9.2)
(9.3)