In this paper we consider a notion of I-Luzin set which generalizes the classical notion of Luzin set and Sierpiński set on Euclidean spaces. We show that there is a translation invariant σ-ideal I with Borel base for which I-Luzin set can be I-measurable. If we additionally assume that I has the Smital property (or its weaker version) then I-Luzin sets are I-nonmeasurable. We give some constructions of I-Luzin sets involving additive structure of R n . Moreover, we show that if c is regular, L is a generalized Luzin set and S is a generalized Sierpiński set then the complex sum L + S belongs to Marczewski ideal s 0 .
In this paper we consider a notion of universal sets for ideals. We show that there exist universal sets of minimal Borel complexity for classic ideals like null subsets of 2 ω and meager subsets of any Polish space, and demonstrate that the existence of such sets is helpful in establishing some facts about the real line in generic extensions. We also construct universal sets for E -the σ-ideal generated by closed null subsets of 2 ω , and for some ideals connected with forcing notions: K σ subsets of ω ω and the Laver ideal. We also consider Fubini products of ideals and show that there are Σ 0 3 universal sets for N ⊗M and M ⊗ N .X, M the σ-ideal of sets of the first category, N the σ-ideal of null subsets 2010 Mathematics Subject Classification: Primary 54H05; Secondary 03E57.
A $$\sigma $$ σ -ideal $$\mathcal {I}$$ I on a Polish group $$(X,+)$$ ( X , + ) has the Smital Property if for every dense set D and a Borel $$\mathcal {I}$$ I -positive set B the algebraic sum $$D+B$$ D + B is a complement of a set from $$\mathcal {I}$$ I . We consider several variants of this property and study their connections with the countable chain condition, maximality and how well they are preserved via Fubini products. In particular we show that there are $$\mathfrak {c}$$ c many maximal invariant $$\sigma $$ σ -ideals with Borel bases on the Cantor space $$2^\omega $$ 2 ω .
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.