### Abstract

We consider a certain class of normalized positive linear functionals on l^{∞} which extend the Cesàro mean. We study the set of its extreme points and it turns out to be the set of linear functionals constructed from free ultrafilters on natural numbers N. Also, regarding them as finitely additive measures defined on all subsets of N, which are often called density measures, we study a certain additivity property of such measures being equivalent to the completeness of the L^{p}-spaces on such measures. Particularly a necessary and sufficient condition for such a density measure to have this property is obtained.

Original language | English |
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Pages (from-to) | 184-203 |

Number of pages | 20 |

Journal | Journal of Number Theory |

Volume | 176 |

DOIs | |

Publication status | Published - 2017 Jul 1 |

### Fingerprint

### Keywords

- Additive property
- Asymptotic density
- Cesàro mean
- Density measure
- Finitely additive measure

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

**Density measures and additive property.** / Kunisada, Ryoichi.

Research output: Contribution to journal › Article

*Journal of Number Theory*, vol. 176, pp. 184-203. https://doi.org/10.1016/j.jnt.2016.12.013

}

TY - JOUR

T1 - Density measures and additive property

AU - Kunisada, Ryoichi

PY - 2017/7/1

Y1 - 2017/7/1

N2 - We consider a certain class of normalized positive linear functionals on l∞ which extend the Cesàro mean. We study the set of its extreme points and it turns out to be the set of linear functionals constructed from free ultrafilters on natural numbers N. Also, regarding them as finitely additive measures defined on all subsets of N, which are often called density measures, we study a certain additivity property of such measures being equivalent to the completeness of the Lp-spaces on such measures. Particularly a necessary and sufficient condition for such a density measure to have this property is obtained.

AB - We consider a certain class of normalized positive linear functionals on l∞ which extend the Cesàro mean. We study the set of its extreme points and it turns out to be the set of linear functionals constructed from free ultrafilters on natural numbers N. Also, regarding them as finitely additive measures defined on all subsets of N, which are often called density measures, we study a certain additivity property of such measures being equivalent to the completeness of the Lp-spaces on such measures. Particularly a necessary and sufficient condition for such a density measure to have this property is obtained.

KW - Additive property

KW - Asymptotic density

KW - Cesàro mean

KW - Density measure

KW - Finitely additive measure

UR - http://www.scopus.com/inward/record.url?scp=85013200709&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85013200709&partnerID=8YFLogxK

U2 - 10.1016/j.jnt.2016.12.013

DO - 10.1016/j.jnt.2016.12.013

M3 - Article

VL - 176

SP - 184

EP - 203

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

ER -