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- Title
Products of K -Analytic Sets in Locally Compact Groups and Kuczma–Ger Classes.
- Authors
Banakh, Taras; Banakh, Iryna; Jabłońska, Eliza
- Abstract
We prove that for any K-analytic subsets A , B of a locally compact group X if the product A B has empty interior (and is meager) in X, then one of the sets A or B can be covered by countably many closed nowhere dense subsets (of Haar measure zero) in X. This implies that a K-analytic subset A of X can be covered by countably many closed Haar-null sets if the set A A A A has an empty interior in X. It also implies that every non-open K-analytic subgroup of a locally compact group X can be covered by countably many closed Haar-null sets in X (for analytic subgroups of the real line this fact was proved by Laczkovich in 1998). Applying this result to the Kuczma–Ger classes, we prove that an additive function f : X → R on a locally compact topological group X is continuous if and only if f is upper bounded on some K-analytic subset A ⊆ X that cannot be covered by countably many closed Haar-null sets.
- Subjects
COMPACT groups; HAAR integral; TOPOLOGICAL groups; ADDITIVE functions
- Publication
Axioms (2075-1680), 2022, Vol 11, Issue 2, p65
- ISSN
2075-1680
- Publication type
Article
- DOI
10.3390/axioms11020065