## 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.

### MSC:

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

### Citations:

Zbl 0832.26010; Zbl 0850.26004; Zbl 1079.26007; Zbl 1275.26019
Full Text: