Generalized Hake property for integrals of Henstock type (Q2018003)

Hake's theorem is a classical result in integration theory, which says that a real function \(f\) is Henstock-Kurzweil (or Perron) integrable over an interval \([a,b]\subset\mathbb R\) if and only if it is Henstock-Kurzweil integrable on each subinterval \([a,c]\) with \(c\in(a,b)\), and if \(\lim_{c\to b-}\int_a^c f\) exists. The goal of the present paper is to show that an analogue of Hake's theorem is available in a much more general situation when we integrate real functions defined on a Hausdorff topological space \(X\) equipped with a regular Borel measure \(\mu\). In order to define the Henstock-Kurzweil integral in this abstract setting, the authors consider a certain family \(\mathcal I\) of closed sets in \(X\) having a finite positive measure; the elements of \(\mathcal I\) are called generalized intervals. Finite unions of nonoverlapping generalized intervals are referred to as figures. The next ingredient is a so-called differential basis \(\mathcal B\), whose elements \(\beta\in\mathcal B\) are systems of pairs \((I,x)\) with \(I\in\mathcal I\) and \(x\in I\). Given a \(\beta\in\mathcal B\), a \(\beta\)-partition of a figure \(L\subset X\) is a finite collection of point-interval pairs \(\pi=(I_i,x_i)_{i=1}^m\subset\beta\), where the intervals do not overlap each other and their union is \(L\). A real function \(f\) defined on a figure \(L\) is said to be Henstock-Kurzweil integrable if there exists a real number \(A\) with the property that for each \(\varepsilon>0\), there exists a \(\beta\in\mathcal B\) such that \[ \left|\sum_{(I,x)\in\pi} f(x)\mu(I)-A\right|<\varepsilon \] holds for each \(\beta\)-partition \(\pi\) of \(L\). In the classical setting when \(X=\mathbb R\) and \(L=[a,b]\subset\mathbb R\), the differential basis \(\mathcal B\) consists of all \(\delta\)-fine partitions of \([a,b]\) for all possible gauges \(\delta:[a,b]\to(0,\infty)\). Other examples of differential bases for \(X=\mathbb R^2\) can be found in [\textit{K. M. Ostaszewski}, Mem. Am. Math. Soc. 353, 106 p. (1986; Zbl 0596.26005)]. After summarizing some basic properties of the above-mentioned integral, the authors proceed to their main result. They consider a closed set \(E\subset X\), a figure \(L\subset X\), and an increasing sequence of figures \(\{F_k\}_{k=1}^\infty\) contained in \(L\) which do not intersect \(E\) and the union of their interiors is \(L\setminus E\). The Hake-type theorem then provides a necessary and sufficient condition for the integrability of a function \(f:L\to\mathbb R\), provided that \(f\) is integrable on \(E\) and on each of the figures \(F_k\).