A Borel linear subspace of R^\omega that cannot be covered by countably many closed Haar-meager sets
DOI:
https://doi.org/10.12775/TMNA.2023.002Keywords
Additive function, mid-convex function, continuity, Haar-null set, Haar-meager set, null-finite set, Haar-thin set, Polish Abelian group, Ger-Kuczma classesAbstract
We prove that the countable product of lines contains a Haar-null Haar-meager Borel linear subspace $L$ that cannot be covered by countably many closed Haar-meager sets. This example is applied to studying the interplay between various classes of ``large'' sets and Kuczma-Ger classes in the topological vector spaces ${\mathbb R}^n$ for $n\le \omega$.References
I. Banakh, T. Banakh and E. Jablońska, Products of K-analytic sets in locally compact groups and Kuczma–Ger classes, Axioms 11 (2022), no. 2, 65.
T. Banakh, S. Gla̧b, E. Jablońska and J. Swaczyna, Haar-I sets: looking at small sets in Polish groups through compact glasses, Dissert. Math. 564 (2021), 105 pp.
T. Banakh and E. Jablońska, Null-finite sets in metric groups and their applications, Israel J. Math. 230 (2019), no. 1, 361–386.
A. Blass, Combinatorial Cardinal Characteristics of the Continuum, Handbook of Set Theory, vols. 1–3, Springer, Dordrecht, 2010, pp. 395–489.
J.P.R. Christensen, On sets of Haar measure zero in abelian Polish groups, Israel J. Math. 13 (1972), 255–260.
U.B. Darji, On Haar meager sets, Topology Appl. 160 (2013), 2396–2400.
L. Dubikajtis, C. Ferens, R. Ger and M. Kuczma, On Mikusiński’s functional equation, Ann. Polon. Math. 28 (1973), 39–47.
P. Erdős, On some properties of Hamel bases, Colloquium Math. 10 (1963), 267–269.
R. Ger and Z. Kominek, Boundedness and continuity of additive and convex functionals, Aequationes Math. 37 (1989), 252–258.
R. Ger and M. Kuczma, On the boundedness and continuity of convex functions and additive functions, Aequationes Math. 4 (1970), 157–162.
E. Jablońska, Some analogies between Haar meager sets and Haar null sets in abelian Polish groups, J. Math. Anal. Appl. 421 (2015), 1479–1486.
A.S. Kechris, Classical Descriptive Set Theory, Springer, New York, 1998.
M. Laczkovich, Analytic subgroups of the reals, Proc. Amer. Math. Soc. 126 (1998), no. 6, 1783–1790.
B.J. Pettis, Remarks on a theorem of E.J. McShane, Proc. Amer. Math. Soc. 2 (1951), 166–171.
H. Steinhaus, Sur les distances des points des ensembles de mesure positive, Fund. Math. 1 (1920), 99–104.
A. Weil, L’intégration dans les groupes topologiques, Actualités Scientifiques et Industrielles 1145, Hermann, 1965.
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Copyright (c) 2024 Taras Banakh, Eliza Jabłońska
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