Original scientific paper

Uniform density $u$ and ${\text{I}}_u$-convergence on a big set

Paweł Barbarski ; Institute of Mathematics, University of Gdańsk, Gdańsk, Poland
Rafał Filipów ; Institute of Mathematics, University of Gdańsk, Gdańsk, Poland
Nikodem Mrożek ; Institute of Mathematics, University of Gdańsk, Gdańsk, Poland
Piotr Szuca ; Institute of Mathematics, University of Gdańsk, Gdańsk, Poland

Abstract

We point out that
${\text{I}}_u$-convergence (where ${\text{I}}_u$ stands for the ideal of uniform density zero sets)
is not equivalent to the convergence on a set from the dual filter.
Moreover, we show that there are bounded sequences which
are not ${\text{I}}_u$-convergent on any set with a positive uniform density,
i.e.~${\text{I}}_u$ does not have the Bolzano-Weierstrass property. We also study relationship
between the Bolzano-Weierstrass property and nonatomic submeasures.
The Borel complexity of the ideal ${\text{I}}_u$ is determined in the last section.

Keywords

uniform statistical density; statistical density; ideal convergence; Bolzano-Weierstrass property; P-ideal; nonatomic submeasure; Borel set

Hrčak ID:

68629

URI

https://hrcak.srce.hr/68629

Publication date:

10.6.2011.

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