The Hake's theorem on metric measure spaces (Q2356312)

The Hake theorem of the title is: If \(f:[0,1]\mapsto \mathbb R\) then \(f\) is Perron integrable, equivalently Denjoy\(^*\) or H-K integrable, if and only if it is integrable over every \( [c,1], 0<c<1,\) and \(\lim_{c\to1}\int_c^1f\) exists. The value of the limit is then \(\int_0^1f\). Extensions of this result to \(\mathbb R^n\) need some ingenuity as basic arguments do not extend readily to the higher dimensional situation. Further the extensions obtained depended on properties of \(\mathbb R^n\); see [\textit{C.-A. Faure} and \textit{J. Mawhin}, Real Anal. Exch. 20, No. 2, 622--630 (1995; Zbl 0832.26010); Rapp., Sémin. Math., Louvain, Nouv. Sér. 237--244, 220--221 (1994; Zbl 0850.26004); \textit{P. Muldowney} and \textit{V. A. Skvortsov}, Math. Notes 78, No. 2, 228--233 (2005); translation from Mat. Zametki 78, No. 2, 251--258 (2005; Zbl 1079.26007); \textit{S. P. Singh} and \textit{I. K. Rana}, Real Anal. Exch. 37, No. 2, 477--488 (2012; Zbl 1275.26019)]. In the present paper, the authors uses properties of the Henstock variational measures to obtain an extension valid in metric measure spaces: Let \(I\) be a closed ball and suppose that for each compact interval \(J\subset I\) with \(J \cap\partial I=\emptyset\) the function\(f\) is integrable over \(J\), \(\int_Jf= F(J)\): then \(f\) is integrable over \(I\), with \(\int_If= F(I)\), if and only if \(V_F(\partial I) = 0\), where \(V_F\) denotes the Henstock variational measure. For full details and some open questions reference should be made to the paper.