A nonabsolute integral on measure spaces that includes the Davies-McShane integral (Q2778268)

In certain metric spaces it is shown that absolute Henstock integrability is equivalent to Davies, Davies-McShane and McShane integrability. NEWLINENEWLINENEWLINELet \((X,d)\) be a metric space with topology \({\mathcal T}\) induced by the metric \(d\) on \(X\) and let \((X,\Omega,\imath)\) be a measure space such that \({\mathcal T}\subset\Omega\). The following inner and outer regularity is assumed: for all measurable \(W\subset\Omega\) and all \(\varepsilon >0\) there is an open set \(U\) and a closed set \(Y\) such that \(Y\subset W\subset U\) and \(\imath (U\setminus Y)<\varepsilon\). It is assumed that the measure of each open ball is positive and that an open ball and its closure have the same measure. Intervals are defined to be finite intersections of closed and scalloped balls. If \(A\) and \(B\) are closed balls such that neither is a subset of the other and their intersection is nonempty then \(A\setminus B\) is a scalloped ball. Elementary sets are finite unions of intervals. The Henstock and McShane integrals are then defined in the usual way for functions mapping from an elementary set to the real line, with partitions being made up of intervals. Existence of \(\delta\)-fine partitions using scalloped balls was proved in [\textit{N. W. Leng} and \textit{L. P. Yee}, Bull. Lond. Math. Soc. 32, No. 1, 34-38 (2000; Zbl 1028.26005)]. NEWLINENEWLINENEWLINESeveral elementary results are proved for absolute Henstock integrals in this setting. Then it is shown that McShane and absolute Henstock integrability are equivalent. By appealing to a result of \textit{R. Henstock} [Real Anal. Exch. 19, No. 1, 121-128 (1994; Zbl 0823.28001)] on the Davies and Davies-McShane integrals it is shown that absolute Henstock integrability is equivalent to Davies, Davies-McShane and McShane integrability. NEWLINENEWLINENEWLINE`Davies' is misspelled as `Davis' several times in the paper.