### 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.

Original language | English |
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Pages (from-to) | 1941-1951 |

Number of pages | 11 |

Journal | Proceedings of the American Mathematical Society |

Volume | 147 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2019 May 1 |

### Fingerprint

### Keywords

- Asymptotic density
- Density measures
- Finitely additive measures

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Proceedings of the American Mathematical Society*,

*147*(5), 1941-1951. https://doi.org/10.1090/proc/14392

**On a relation between density measures and a certain flow.** / Kunisada, Ryoichi; Shah, Nimish.

Research output: Contribution to journal › Article

*Proceedings of the American Mathematical Society*, vol. 147, no. 5, pp. 1941-1951. https://doi.org/10.1090/proc/14392

}

TY - JOUR

T1 - On a relation between density measures and a certain flow

AU - Kunisada, Ryoichi

AU - Shah, Nimish

PY - 2019/5/1

Y1 - 2019/5/1

N2 - 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.

AB - 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.

KW - Asymptotic density

KW - Density measures

KW - Finitely additive measures

UR - http://www.scopus.com/inward/record.url?scp=85065490109&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85065490109&partnerID=8YFLogxK

U2 - 10.1090/proc/14392

DO - 10.1090/proc/14392

M3 - Article

AN - SCOPUS:85065490109

VL - 147

SP - 1941

EP - 1951

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 5

ER -