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

      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

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