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NONMEASURABLE SETS AND UNIONS WITH RESPECT TO TREE IDEALS

Part of: Classical measure theory Set theory

Published online by Cambridge University Press:  19 June 2020

MARCIN MICHALSKI ,
ROBERT RAŁOWSKI  and
SZYMON ŻEBERSKI
MARCIN MICHALSKI
Affiliation:
DEPARTMENT OF FUNDAMENTALS OF COMPUTER SCIENCE FACULTY OF FUNDAMENTAL PROBLEMS OF TECHNOLOGY WROCŁAW UNIVERSITY OF SCIENCE AND TECHNOLOGY WYBRZEŻE WYSPIAŃSKIEGO 27, 50-370 WROCŁAW, POLAND E-mail: marcin.k.michalski@pwr.edu.pl E-mail: robert.ralowski@pwr.edu.pl E-mail: szymon.zeberski@pwr.edu.pl
ROBERT RAŁOWSKI
Affiliation:
DEPARTMENT OF FUNDAMENTALS OF COMPUTER SCIENCE FACULTY OF FUNDAMENTAL PROBLEMS OF TECHNOLOGY WROCŁAW UNIVERSITY OF SCIENCE AND TECHNOLOGY WYBRZEŻE WYSPIAŃSKIEGO 27, 50-370 WROCŁAW, POLAND E-mail: marcin.k.michalski@pwr.edu.pl E-mail: robert.ralowski@pwr.edu.pl E-mail: szymon.zeberski@pwr.edu.pl
SZYMON ŻEBERSKI
Affiliation:
DEPARTMENT OF FUNDAMENTALS OF COMPUTER SCIENCE FACULTY OF FUNDAMENTAL PROBLEMS OF TECHNOLOGY WROCŁAW UNIVERSITY OF SCIENCE AND TECHNOLOGY WYBRZEŻE WYSPIAŃSKIEGO 27, 50-370 WROCŁAW, POLAND E-mail: marcin.k.michalski@pwr.edu.pl E-mail: robert.ralowski@pwr.edu.pl E-mail: szymon.zeberski@pwr.edu.pl
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Abstract

In this paper, we consider a notion of nonmeasurablity with respect to Marczewski and Marczewski-like tree ideals $s_0$ , $m_0$ , $l_0$ , $cl_0$ , $h_0,$ and $ch_0$ . We show that there exists a subset of the Baire space $\omega ^\omega ,$ which is s-, l-, and m-nonmeasurable that forms a dominating m.e.d. family. We investigate a notion of ${\mathbb {T}}$ -Bernstein sets—sets which intersect but do not contain any body of any tree from a given family of trees ${\mathbb {T}}$ . We also obtain a result on ${\mathcal {I}}$ -Luzin sets, namely, we prove that if ${\mathfrak {c}}$ is a regular cardinal, then the algebraic sum (considered on the real line ${\mathbb {R}}$ ) of a generalized Luzin set and a generalized Sierpiński set belongs to $s_0, m_0$ , $l_0,$ and $cl_0$ .

Keywords

Marczewski idealperfect treeLaver treeMiller treeHechler treeBernstein setm.e.d. familydominating familynonmeasurable setPolish spaceContinuum Hypothesistree idealI-Luzin set

MSC classification

Primary: 03E17: Cardinal characteristics of the continuum 03E50: Continuum hypothesis and Martin's axiom 03E75: Applications of set theory
Secondary: 28A99: None of the above, but in this section
Type
Articles
Information
Bulletin of Symbolic Logic , Volume 26 , Issue 1 , March 2020 , pp. 1 - 14
Copyright
© The Association for Symbolic Logic 2020

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