6 - Extending the integral

Summary

In Chapter 5 we have demonstrated that the R-integral is an averaging process of appreciable appeal. At the same time, the additivity property of the R-integral is genuinely restricted: cf. Proposition 5.1.8 with Remark 6.1.2, (4) below. To correct this deficiency, we follow ideas of Mařík and extend the R-integral by a transfinite iteration of improper integrals — a process similar to Cauchy's extensions in the constructive definition of the Denjoy integral [75, Chapter 8, Section 4]. We show that the extended integral inherits all desirable properties of the R-integral, and that it is additive in the usual way. The extension is maximal in the sense that the extended integral is closed with respect to further formations of improper integrals.

Buczolich's example

We present an example, constructed by Buczolich [12], which implies that the assumptions of Propositions 3.6.4, (ii) and 5.1.8 are essential.

Proposition 6.1.1.Assume m = 2, and let K:= [0, l]². There are a vector field υ ∈ C(ℝ^m; ℝ^m), whose flux is denoted by H, an open BV set W ⊂ K, and a BV set A ⊂ W having the following properties.