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

      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,

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