Absolute integration using Vitali covers (Q1308837)

A collection \(\beta\) of interval-point pairs is said to be a Vitali cover of a set \(E\subset \mathbb{R}\) if for each \(\varepsilon>0\) and any \(x \in E\) there is an interval-point pair \((I,x) \in \beta\) such that \(x \in I\) and the length \(| I |\) of \(I\) is less than \(\varepsilon\). \(D=\bigl\{ (I,x) \bigr\} \subset \beta\) is said to be a partial \(\beta\)-partition of \([a,b]\) if \(\{I:(I,x) \in D\}\) is a finite collection of nonoverlapping subintervals of \([a,b]\). Let \({\mathcal B}\) be a collection of Vitali covers on \([a,b]\). If \(\beta \in{\mathcal B}\), in analogy with the Kurzweil-Henstock integral, the authors say that a real function \(f\) defined on \([a,b]\) is \({\mathcal B}^*\) integrable if there is some real \(A\) with the property that, for every \(\varepsilon>0\) there exists \(\eta>0\) and \(\beta \in{\mathcal B}\) such that for any partial \(\beta\)-partition \(D=\bigl\{ (I,x) \bigr\}\) of \([a,b]\) with \((D) \sum | I |>b-a-\eta\), we have \(\bigl|(D)\sum f(x) \bigr| I | -A |<\varepsilon\). Various properties (uniqueness, Cauchy criterion, Saks-Henstock lemma) are proved and its relations with the Lebesgue and MacShane's nonstochastic Itô- belated integrals are discussed.