The Hake’s theorem on metric measure spaces. (English) Zbl 1326.26017

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 [C.-A. Faure and 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); P. Muldowney and V. A. Skvortsov, Math. Notes 78, No. 2, 228–233 (2005); translation from Mat. Zametki 78, No. 2, 251–258 (2005; Zbl 1079.26007); S. P. Singh and 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.


26A39 Denjoy and Perron integrals, other special integrals
46G99 Measures, integration, derivative, holomorphy (all involving infinite-dimensional spaces)
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