On comparable and non-comparable Hausdorff measures

Authors

  • M.V. Lebid
  • G.M. Torbin

DOI:

https://doi.org/10.15407/dopovidi2014.08.035

Keywords:

Hausdorff measures, theory of fractals, triviality

Abstract

General methods of calculation of the Hausdorff–Besicovitch dimension are developed, and the triviality (non-triviality) of net Hausdorff measures is studied. Sufficient conditions for a fine covering family to generate comparable net Hausdorff measures Hα (·, Φ) are given. General sufficient conditions for a fine covering family to be faithful are also found. The faithfulness for the family of coverings generated by the factorial expansion for real numbers is proved. It is shown that the family of faithful coverings is essentially wider than the family of coverings generating the comparable net Hausdorff measures.

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References

Beresnevich V., Velani S. Ann. of Math., 2006, 164: 971–992. https://doi.org/10.4007/annals.2006.164.971

Zhou Z., Feng L. Nonlinearity, 2004, 17, No 2: 493–502. https://doi.org/10.1088/0951-7715/17/2/007

Baranski K. Adv. Math., 2007, 210: 215–245. https://doi.org/10.1016/j.aim.2006.06.005

Rogers C. Hausdorff measures, London: Cambridge Univ. Press, 1970.

Besicovitch A. Indag. Math., 1952, 14: 339–344. https://doi.org/10.1016/S1385-7258(52)50045-3

Billingsley P. Ill. J. Math., 1961, 5: 291–198.

Turbin A. F., Pratsevityy N. V. Fractal sets, functions, distributions, Kyiv: Nauk. dumka, 1992 (in Russian).

Albeverio S., Torbin G. Bull. Sci. Math., 2005, 129, No 4: 356–367. https://doi.org/10.1016/j.bulsci.2004.12.001

Cutler C. Internat. J. Math. and Math. Sci., 1988, 2, No 4: 643–650. https://doi.org/10.1155/S016117128800078X

Pratsiovytyi M. V., Torbin G. M. Nauk. zap. NPU im. M. P. Dragomanova. Fiz.-mat. nauky, 2003, 4, No 1: 207–215 (in Ukrainian).

Albeverio S., Koshmanenko V., Pratsiovytyi M., Torbin G. Methods of Functional Analysis and Topology, 2011, 17, No 2: 97–111.

Bernardi M., Bondioli C. J. Anal. and its Appl., 1999, 18, No 3: 733–751.

Erdos P., Renyi A. Acta Math. Acad. Sci. Hungar, 1959, 10: 207–215.

Albeverio S., Pratsiovytyi M., Torbin G. Ergodic Theory and Dynamical Systems, 2004, 24, No 1: 1–16. https://doi.org/10.1017/S0143385703000397

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Published

01.03.2025

How to Cite

Lebid, M., & Torbin, G. (2025). On comparable and non-comparable Hausdorff measures . Reports of the National Academy of Sciences of Ukraine, (8), 35–40. https://doi.org/10.15407/dopovidi2014.08.035

Issue

No. 8 (2014): Reports of the National Academy of Sciences of Ukraine

Section

Mathematics

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