×

Recovering measures from approximate values on balls. (English) Zbl 1351.28028

The authors consider the problem of reconstructing a measure on a metric space from its values on the family of closed balls. The first result shows that this is possible for Borel measures on separable metric spaces that satisfy Besicovitch’s theorem on covering set, up to measure zero, by disjoint families of balls. The second main result gives sufficient conditions under which one can get a measure that differs from the given measure by (at most) a multiplicative constant when one knows the measures of the balls, again up to at most that constant.
Reviewer: K. P. Hart (Delft)

MSC:

28C15 Set functions and measures on topological spaces (regularity of measures, etc.)
28A78 Hausdorff and packing measures
49Q15 Geometric measure and integration theory, integral and normal currents in optimization
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Bruckner, A. M., J. B. Bruckner, and B. S. Thomson: Elementary real analysis. - Prentice Hall, 2001. · Zbl 1018.01004
[2] Buet, B.: Approximation de surfaces par des varifolds discrets : représentation, courbure, rectifiabilité. - PhD thesis, Université Claude Bernard Lyon 1, 2014.
[3] Buet, B.: Quantitative conditions of rectifiability for varifolds. - Ann. Inst. Fourier (Grenoble) 65:6, 2015, 2449–2506. · Zbl 1344.49071
[4] Buet, B., G. P. Leonardi, and S. Masnou: Surface approximation, discrete varifolds, and regularized first variation. - In preparation. · Zbl 1378.49052
[5] Davies, R. O.: Measures not approximable or not specifiable by means of balls. - Mathematika 18, 1971, 157–160. · Zbl 0229.28005
[6] Edgar, G. A.: Packing measure in general metric space. - Real Anal. Exchange 26:2, 2000, 831–852. · Zbl 1010.28007
[7] Evans, L. C., and R. F. Gariepy: Measure theory and fine properties of functions. - Stud. Adv. Math., CRC Press, Boca Raton, FL, 1992. · Zbl 0804.28001
[8] Federer, H.: Geometric measure theory. - Grundlehren Math. Wiss. 153, Springer-Verlag, New York, 1969. · Zbl 0176.00801
[9] Preiss, D.: Geometric measure theory in Banach spaces. - North-Holland Publishing Co., Amsterdam, 2001. · Zbl 1064.46068
[10] Preiss, D., and J. Tišer: Measures in Banach spaces are determined by their values on balls. - Mathematika 38:2, 1991, 391–397. · Zbl 0755.28006
[11] Taylor, S. J., and C. Tricot: Packing measure, and its evaluation for a Brownian path. Trans. Amer. Math. Soc. 288:2, 1985, 679–699. · Zbl 0537.28003
[12] Thomson, B. S.: Construction of measures in metric spaces. - J. London Math. Soc. 14:2, 1976, 21–26. · Zbl 0348.28004
[13] Tricot, C.: Two definitions of fractional dimension. - Math. Proc. Cambridge Philos. Soc. 91:1, 1982, 57–74. Received 24 December 2015 Accepted 31 March 2016 · Zbl 0483.28010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.