target="_blank" rel="nofollow" href="#fb3_img_img_b07cbf0a-311f-545c-a19b-c10bb6e04b00.png" alt="upper X"/> 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
