Generalized Ordinary Differential Equations in Abstract Spaces and Applications. Группа авторов

In order to prove that , it is enough to write the usual Riemannian‐type sum as two sums (with the positive and negative parts of the sum):

      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 2 they 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