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ON SEQUENCES OF HOMOMORPHISMS INTO MEASURE ALGEBRAS AND THE EFIMOV PROBLEM

Part of: Classical measure theory Miscellaneous topics in measure theory Set theory

Published online by Cambridge University Press:  13 September 2021

PIOTR BORODULIN–NADZIEJA  and
DAMIAN SOBOTA
PIOTR BORODULIN–NADZIEJA
Affiliation:
INSTYTUT MATEMATYCZNY UNIWERSYTET WROCŁAWSKI PL. GRUNWALDZKI 2/4, 50-384 WROCŁAW, POLAND E-mail: pborod@math.uni.wroc.pl
DAMIAN SOBOTA*
Affiliation:
KURT GÖDEL RESEARCH CENTER FOR MATHEMATICAL LOGIC DEPARTMENT OF MATHEMATICS UNIVERSITY OF VIENNA KOLINGASSE 14-16, 1090 VIENNA, AUSTRIA
*
E-mail: damian.sobota@univie.ac.at
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Abstract

For given Boolean algebras $\mathbb {A}$ and $\mathbb {B}$ we endow the space $\mathcal {H}(\mathbb {A},\mathbb {B})$ of all Boolean homomorphisms from $\mathbb {A}$ to $\mathbb {B}$ with various topologies and study convergence properties of sequences in $\mathcal {H}(\mathbb {A},\mathbb {B})$ . We are in particular interested in the situation when $\mathbb {B}$ is a measure algebra as in this case we obtain a natural tool for studying topological convergence properties of sequences of ultrafilters on $\mathbb {A}$ in random extensions of the set-theoretical universe. This appears to have strong connections with Dow and Fremlin’s result stating that there are Efimov spaces in the random model. We also investigate relations between topologies on $\mathcal {H}(\mathbb {A},\mathbb {B})$ for a Boolean algebra $\mathbb {B}$ carrying a strictly positive measure and convergence properties of sequences of measures on $\mathbb {A}$ .

Keywords

Efimov problemmeasure algebrasweak convergence of measuresalgebraic convergencerandom forcingperfect Hamming codes

MSC classification

Primary: 28A20: Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
Secondary: 28A60: Measures on Boolean rings, measure algebras 03E40: Other aspects of forcing and Boolean-valued models 28E15: Other connections with logic and set theory 03E75: Applications of set theory 28A33: Spaces of measures, convergence of measures
Type
Article
Information
The Journal of Symbolic Logic , Volume 88 , Issue 1 , March 2023 , pp. 191 - 218
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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