Generalized Ordinary Differential Equations in Abstract Spaces and Applications

for every ‐fine corresponding to a given .

As we already mentioned, we will consider a more general definition of the Kurzweil integral in Chapter 2. Thus, in the remaining of this chapter, we refer to the integrals

as Perron–Stieltjes integrals, where and .

As it should be expected, the above integrals are linear and additive over nonoverlapping intervals. These facts will be put aside for a while, because in Chapter 2they will be proved for the more general form of the Kurzweil integral. In the meantime, we present a simple example of a function which is Riemann improper integrable (and, hence, also Perron integrable, due to Theorem 2.9), but it is not Lebesgue integrable (because it is not absolutely integrable).

Example 1.43:Let be given by , for , and , for some Then, it is not difficult to prove that exists, but , once

Another example is also needed at this point. Borrowed from [73, example 2.1], the example below exhibits a function (that is, belongs to , but not to ), satisfying . However, for almost every . Thus, such a function is also an element of (because it belongs to ) which does not belong to , showing that, in the infinite dimensional‐valued case, may be a proper subset of .

Example 1.44:Let be an arbitrary set and let be a normed space. A family of elements of is summable with sum (we write ), if for every , there is a finite subset such that for every finite subset with ,