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

which are Lebesgue integrable with finite integral. See the appendix of this chapter. As a matter of fact, the Fundamental Theorem of Calculus for the Henstock integral (see Theorem 1.73) yields that . Optionally, one can verify that simply by noticing that

      for every .

      Claim. , that is, is Riemann–McShane integrable (see the appendix of this chapter).

      Is is sufficient to prove that, given , we can find such that for every -fine (the reader may want to check the notation in the appendix of this chapter),

Consider and take a -fine .

       If and , then . Therefore, and, hence,

       If and , then . Therefore,and we obtain

      Finally, we get

      and the Claim is proved.

      A less restrict version of the Fundamental Theorem of Calculus is stated next. A proof of it follows as in [108, Theorem 9.6].

      Theorem 1.75 (Fundamental Theorem of Calculus): Suppose is a continuous function such that there exists the derivative , for nearly everywhere on i.e. except for a countable subset of . Then, and

      Now, we present a class of functions , laying between absolute continuous and continuous functions, for which we can obtain a version of the Fundamental Theorem of Calculus for Henstock vector integrals. Let denote the Lebesgue measure.

Definition 1.76: A function satisfies the strong Lusin condition, and we write , if given and with , then there is a gauge on such that for every -fine with for all