The Kurzweil-Henstock Integral for Undergraduates by Alessandro Fonda

Author:Alessandro Fonda
Language: eng
Format: epub
ISBN: 9783319953212
Publisher: Springer International Publishing

is a diffeomorphism of one variable between the open sets and sections of U and V, respectively. Using the Reduction Theorem and the one-dimensional formula of change of variables proved above, we have that

Hence, being φ = β ∘ α, we have:

We have then proved that, for every u ∈ A, there is a δ(u) > 0 such that the thesis holds true when D is contained in B[u, δ(u)]. A gauge δ is thus defined on A. By Lemma 2.20, we can now cover A with a countable family (J k) of non-overlapping rectangles, each contained in a rectangle of the type B[u, δ(u)], so that the formula holds for the closed sets contained in any of these rectangles.

At this point let us consider an arbitrary closed subset D of A. Then, the formula holds for each D ∩ J k and, by the complete additivity of the integral and the fact that the sets φ(D ∩ J k) are non-overlapping (as a consequence of Lemma 2.36), we have:

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