On a relation between density measures and a certain flow
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- by Ryoichi Kunisada PDF
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Abstract:
We study extensions of the asymptotic density to a finitely additive measure defined on all subsets of natural numbers. Such measures are called density measures. We consider a class of density measures constructed from free ultrafilters on natural numbers and investigate absolute continuity and singularity for those density measures. In particular, for any pair of such density measures we prove necessary and sufficient conditions that one is absolutely continuous with respect to the other and that they are singular. Also we prove similar results for weak absolute continuity and strong singularity. These results are formulated in terms of topological dynamics.References
- Achille Basile and K. P. S. Bhaskara Rao, Completeness of $L_p$-spaces in the finitely additive setting and related stories, J. Math. Anal. Appl. 248 (2000), no. 2, 588–624. MR 1776031, DOI 10.1006/jmaa.2000.6946
- Wayne C. Bell and John W. Hagood, Completeness of $\scr L^p$ spaces and Radon-Nikodym theorems for unbounded finitely additive measures, J. Math. Anal. Appl. 218 (1998), no. 1, 82–96. MR 1601849, DOI 10.1006/jmaa.1997.5744
- K. P. S. Bhaskara Rao and M. Bhaskara Rao, Theory of charges, Pure and Applied Mathematics, vol. 109, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. A study of finitely additive measures; With a foreword by D. M. Stone. MR 751777
- A. Blass, R. Frankiewicz, G. Plebanek, and C. Ryll-Nardzewski, A note on extensions of asymptotic density, Proc. Amer. Math. Soc. 129 (2001), no. 11, 3313–3320. MR 1845008, DOI 10.1090/S0002-9939-01-05941-X
- Karel Hrbacek and Thomas Jech, Introduction to set theory, 3rd ed., Monographs and Textbooks in Pure and Applied Mathematics, vol. 220, Marcel Dekker, Inc., New York, 1999. MR 1697766
- Ryoichi Kunisada, Density measures and additive property, J. Number Theory 176 (2017), 184–203. MR 3622126, DOI 10.1016/j.jnt.2016.12.013
- Alan H. Mekler, Finitely additive measures on ${\rm \textbf {N}}$ and the additive property, Proc. Amer. Math. Soc. 92 (1984), no. 3, 439–444. MR 759670, DOI 10.1090/S0002-9939-1984-0759670-1
Additional Information
- Ryoichi Kunisada
- Affiliation: Faculty of Education and Integrated Arts and Science, Waseda University, Shinjuku-ku, Tokyo 169-8050, Japan
- MR Author ID: 1201850
- Email: tk-waseda@ruri.waseda.jp
- Received by editor(s): March 10, 2017
- Received by editor(s) in revised form: July 12, 2018, and September 2, 2018
- Published electronically: February 6, 2019
- Communicated by: Nimish Shah
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 1941-1951
- MSC (2010): Primary 11B05; Secondary 46E27
- DOI: https://doi.org/10.1090/proc/14392
- MathSciNet review: 3937672