Haar-I sets: Looking at small sets in Polish groups through compact glasses

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F21%3A00546269" target="_blank" >RIV/67985840:_____/21:00546269 - isvavai.cz</a>

Výsledek na webu

<a href="http://dx.doi.org/10.4064/dm812-2-2021" target="_blank" >http://dx.doi.org/10.4064/dm812-2-2021</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.4064/dm812-2-2021" target="_blank" >10.4064/dm812-2-2021</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Haar-I sets: Looking at small sets in Polish groups through compact glasses

Popis výsledku v původním jazyce

Generalizing Christensen’s notion of a Haar-null set and Darji’s notion of a Haar-meager set, we introduce and study the notion of a Haar-I set in a Polish group. Here I is an ideal of subsets of some compact metrizable space K. A Borel subset B ⊆ X of a Polish group X is called Haar-I if there exists a continuous map f: K → X such that f−1 (B + x) ∈ I for all x ∈ X. Moreover, B is generically Haar-I if the set of witness functions {f ∈ C(K, X): ∀x ∈ X f−1 (B +x) ∈ I} is comeager in the function space C(K, X). We study (generically) Haar-I sets in Polish groups for many concrete and abstract ideals I, and construct the corresponding distinguishing examples. We prove some results on Borel hulls of Haar-I sets, generalizing results of Solecki, Elekes, Vidnyánszky, Doležal, Vlasák on Borel hulls of Haar-null and Haar-meager sets. We also establish various Steinhaus properties of the families of (generically) Haar-I sets in Polish groups for various ideals I.

Název v anglickém jazyce

Haar-I sets: Looking at small sets in Polish groups through compact glasses

Popis výsledku anglicky

Generalizing Christensen’s notion of a Haar-null set and Darji’s notion of a Haar-meager set, we introduce and study the notion of a Haar-I set in a Polish group. Here I is an ideal of subsets of some compact metrizable space K. A Borel subset B ⊆ X of a Polish group X is called Haar-I if there exists a continuous map f: K → X such that f−1 (B + x) ∈ I for all x ∈ X. Moreover, B is generically Haar-I if the set of witness functions {f ∈ C(K, X): ∀x ∈ X f−1 (B +x) ∈ I} is comeager in the function space C(K, X). We study (generically) Haar-I sets in Polish groups for many concrete and abstract ideals I, and construct the corresponding distinguishing examples. We prove some results on Borel hulls of Haar-I sets, generalizing results of Solecki, Elekes, Vidnyánszky, Doležal, Vlasák on Borel hulls of Haar-null and Haar-meager sets. We also establish various Steinhaus properties of the families of (generically) Haar-I sets in Polish groups for various ideals I.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2021

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Dissertationes Mathematicae

ISSN

0012-3862

e-ISSN

1730-6310

Svazek periodika

564

Číslo periodika v rámci svazku

August

Stát vydavatele periodika

PL - Polská republika

Počet stran výsledku

105

Strana od-do

1-105

Kód UT WoS článku

000697721200001

EID výsledku v databázi Scopus

2-s2.0-85100276348