Generalized Ordinary Differential Equations in Abstract Spaces and Applications

Example 1.92tells us that the elements of are not absolutely integrable.

When McShane's idea is applied to Kurzweil and Henstock vector integrals, the story changes. In fact, the modification of McShane applied to the Kurzweil vector integral originates an integral which encompasses the Bochner–Lebesgue integral (see Example 1.74). On the other hand, when McShane's idea is used to modify the variational Henstock integral, we obtain exactly the Bochner–Lebesgue integral (see [[47] and [131]]). Thus, if denotes the space of Henstock–McShane integrable functions , that is, is integrable in the sense of Henstock with the modification of McShane, then

We will prove this equality in the sequel. Furthermore, , and , where we use the notation and to denote, respectively, the spaces of Kurzweil–McShane and Riemann–McShane integrable functions from to . For other interesting results, the reader may want to consult [55].

Our aim in the remaining of this chapter is to show that the integrals of Bochner–Lebesgue and Henstock–McShane coincide. See [132, Theorem 10.4]. The next results are due to C. S. Hönig. They belong to a brochure of a series of lectures Professor Hönig gave in Rio de Janeiro in 1993. We include the proofs here, once the brochure is in Portuguese.

Lemma 1.93: Let be a sequence in and be a function. Suppose there exists

Then, and

Proof. Given , take such that for ,

and take a gauge on such that for every -fine ,

(1.A.2)

The limit exists, since for ,

Hence, if , then

Thus, the first summand on the right-hand side of the last inequality is smaller than by ( 1.A.2), the third summand is smaller than by the definition of and, if we refine the gauge , we may suppose, by the definition of , that the second summand is smaller than , and the proof is complete.

We show next that Lemma 1.93remains valid if we replace by , that is, if instead of the space of Kurzweil–McSchane integrable functions, we consider the space of Henstock–McSchane integrable functions.