On comparable and non-comparable Hausdorff measures

TitleOn comparable and non-comparable Hausdorff measures
Publication TypeJournal Article
Year of Publication2014
AuthorsLebid, MV, Torbin, GM
Abbreviated Key TitleDopov. Nac. akad. nauk Ukr.
Date Published8/2014

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.

KeywordsHausdorff measures, theory of fractals, triviality

1. Beresnevich V., Velani S. Ann. of Math., 2006, 164: 971–992. https://doi.org/10.4007/annals.2006.164.971
2. Zhou Z., Feng L. Nonlinearity, 2004, 17, No 2: 493–502. https://doi.org/10.1088/0951-7715/17/2/007
3. Baranski K. Adv. Math., 2007, 210: 215–245. https://doi.org/10.1016/j.aim.2006.06.005
4. Rogers C. Hausdorff measures, London: Cambridge Univ. Press, 1970.
5. Besicovitch A. Indag. Math., 1952, 14: 339–344. https://doi.org/10.1016/S1385-7258(52)50045-3
6. Billingsley P. Ill. J. Math., 1961, 5: 291–198.
7. Turbin A. F., Pratsevityy N. V. Fractal sets, functions, distributions, Kyiv: Nauk. dumka, 1992 (in Russian).
8. Albeverio S., Torbin G. Bull. Sci. Math., 2005, 129, No 4: 356–367. https://doi.org/10.1016/j.bulsci.2004.12.001
9. Cutler C. Internat. J. Math. and Math. Sci., 1988, 2, No 4: 643–650. https://doi.org/10.1155/S016117128800078X
10. Pratsiovytyi M. V., Torbin G. M. Nauk. zap. NPU im. M. P. Dragomanova. Fiz.-mat. nauky, 2003, 4, No 1: 207–215 (in Ukrainian).
11. Albeverio S., Koshmanenko V., Pratsiovytyi M., Torbin G. Methods of Functional Analysis and Topology, 2011, 17, No 2: 97–111.
12. Bernardi M., Bondioli C. J. Anal. and its Appl., 1999, 18, No 3: 733–751.
13. Erdos P., Renyi A. Acta Math. Acad. Sci. Hungar, 1959, 10: 207–215.
14. Albeverio S., Pratsiovytyi M., Torbin G. Ergodic Theory and Dynamical Systems, 2004, 24, No 1: 1–16. https://doi.org/10.1017/S0143385703000397