On Banach and Kuratowski Theorem and generalized strong sequences
Abstract
In 1929, Banach and Kuratowski proved under CH a combinatorial theorem, which implies that there is not a non-vanishing σ-additive finite measure on ℝ which is defined for every set of reals. In 2003 Bartoszyński and Halbeisen proved that Banach and Kuratowski theorem is equivalent to the existence of a K-Lusin set of the cardinality continuum an the existence of such sets is independent of ZFC + ¬ CH. On the other hand in 1965 Efimov introduced the strong sequences method which used to prove some well-known theorems in dyadic spaces. The aim of this paper is to show that all this notions can be generalized, i.e. considered in the generalized Baire space and to show some equivalences among them. Moreover, some applications in the direction of calibres, boundedness and partitions relations are also shown.
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BibTeX
@misc{jureczko2026banach,
title = {{On Banach and Kuratowski Theorem and generalized strong sequences}},
author = {Joanna Jureczko},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13887},
eprint = {2609.13887},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13887},
}
RIS
TY - UNPB
TI - On Banach and Kuratowski Theorem and generalized strong sequences
AU - Joanna Jureczko
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.13887
UR - https://arxiv.org/abs/2609.13887
AB - In 1929, Banach and Kuratowski proved under CH a combinatorial theorem, which implies that there is not a non-vanishing σ-additive finite measure on ℝ which is defined for every set of reals. In 2003 Bartoszyński and Halbeisen proved that Banach and Kuratowski theorem is equivalent to the existence of a K-Lusin set of the cardinality continuum an the existence of such sets is independent of ZFC + ¬ CH. On the other hand in 1965 Efimov introduced the strong sequences method which used to prove some well-known theorems in dyadic spaces. The aim of this paper is to show that all this notions can be generalized, i.e. considered in the generalized Baire space and to show some equivalences among them. Moreover, some applications in the direction of calibres, boundedness and partitions relations are also shown.
ER -