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

Rana, Inder K.; Singh, Surinder Pal

Real Anal. Exch. 39(2013-2014), No. 2, 447-458 (2014).

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.

Reviewer: Peter S. Bullen (Vancouver)

MSC:

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

Keywords:

Henstock-Kurzweil integral; variational measures; Hake’s theorem

Citations:

Zbl 0832.26010; Zbl 0850.26004; Zbl 1079.26007; Zbl 1275.26019