Generalized Ordinary Differential Equations in Abstract Spaces and Applications

and, hence,

Thus, is absolutely continuous.

On the other hand, is nowhere differentiable (see [73], Example 3.1). Then, the Lebesgue theorem implies , where by , we denote the space of functions from to 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 , we have