The Nikodym property and cardinal characteristics of the continuum

Damian Sobota

doi:10.1016/j.apal.2018.07.003

The Nikodym property and cardinal characteristics of the continuum

Damian Sobota

Publications: Contribution to journal › Article › Peer Reviewed

Abstract

We present a general method of constructing Boolean algebras with the Nikodym property and of some given cardinalities. The construction is dependent on the values of some classical cardinal characteristics of the continuum. As a result we obtain a consistent example of an infinite Boolean algebra with the Nikodym property and of cardinality strictly less than the continuum c. It follows that the existence of such an algebra is undecidable by the usual axioms of set theory. Besides, our results shed some new light on the Efimov problem and cofinalities of Boolean algebras.

Original language	English
Pages (from-to)	1-35
Number of pages	35
Journal	Annals of Pure and Applied Logic
Volume	170
Issue number	1
DOIs	https://doi.org/10.1016/j.apal.2018.07.003
Publication status	Published - 2019

Austrian Fields of Science 2012

101032 Functional analysis
101013 Mathematical logic

Keywords

BOOLEAN-ALGEBRAS
CONVERGENCE
Cardinal characteristics of the continuum
Cofinality of Boolean algebras
Efimov problem
GROTHENDIECK
Measures on Boolean algebras
Nikodym Boundedness Theorem
Nikodym property
SPACES
THEOREM

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10.1016/j.apal.2018.07.003

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title = "The Nikodym property and cardinal characteristics of the continuum",

abstract = "We present a general method of constructing Boolean algebras with the Nikodym property and of some given cardinalities. The construction is dependent on the values of some classical cardinal characteristics of the continuum. As a result we obtain a consistent example of an infinite Boolean algebra with the Nikodym property and of cardinality strictly less than the continuum c. It follows that the existence of such an algebra is undecidable by the usual axioms of set theory. Besides, our results shed some new light on the Efimov problem and cofinalities of Boolean algebras.",

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AB - We present a general method of constructing Boolean algebras with the Nikodym property and of some given cardinalities. The construction is dependent on the values of some classical cardinal characteristics of the continuum. As a result we obtain a consistent example of an infinite Boolean algebra with the Nikodym property and of cardinality strictly less than the continuum c. It follows that the existence of such an algebra is undecidable by the usual axioms of set theory. Besides, our results shed some new light on the Efimov problem and cofinalities of Boolean algebras.

KW - BOOLEAN-ALGEBRAS

KW - CONVERGENCE

KW - Cardinal characteristics of the continuum

KW - Cofinality of Boolean algebras

KW - Efimov problem

KW - GROTHENDIECK

KW - Measures on Boolean algebras

KW - Nikodym Boundedness Theorem

KW - Nikodym property

KW - SPACES

KW - THEOREM

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DO - 10.1016/j.apal.2018.07.003

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SN - 0168-0072

VL - 170

SP - 1

EP - 35

JO - Annals of Pure and Applied Logic

JF - Annals of Pure and Applied Logic

IS - 1

ER -

The Nikodym property and cardinal characteristics of the continuum

Abstract

Austrian Fields of Science 2012

Keywords

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