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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: [email protected] E-mail: [email protected] E-mail: [email protected]
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: [email protected] E-mail: [email protected] E-mail: [email protected]
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: [email protected] E-mail: [email protected] E-mail: [email protected]
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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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References

REFERENCES

Brendle, J., Strolling trough paradise . Fundamenta Mathematicae , vol. 148 (1995), pp. 1–25.CrossRefGoogle Scholar
Kechris, A., Classical Descriptive Set Theory , Springer, New York, 2019.Google Scholar
Kunen, K., Set Theory. An Introduction to Independence Proofs , North-Holland, Amsterdam, 1980.Google Scholar
Marczewski (Szpilrajn), E., Remarques sur les fonctions complètement additives d’ensemble et sur les ensembles jouissant de la propriété de Baire . Fundamenta Mathematicae , vol. 22 (1934), pp. 303–311.Google Scholar
Marczewski (Szpilrajn), E., Sur une classe de fonctions de W. Sierpiński et la classe correspondante d’ensembles . Fundamenta Mathematicae , vol. 24 (1935), pp. 17–34.Google Scholar
Michalski, M. and Żeberski, Sz., Some properties of I-luzin . Topology and its Applications , vol. 189 (2015), pp. 122–135.CrossRefGoogle Scholar
Miller, A. W., Hechler and Laver trees. Preprint, 2012, arXiv:1204.5198.Google Scholar
Rałowski, R., Families of sets with nonmeasurable unions with respect to ideals defined by trees . Archive for Mathematical Logic , vol. 54 (2015), pp. 649–658.CrossRefGoogle Scholar
Rałowski, R., Dominating m.a.d. families in Baire space. RIMS Kôkyûroku No. 1949 , 2015, pp. 73–80.Google Scholar
Repický, M., Perfect sets and collapsing continuum . Commentationes Mathematicae Universitatis Carolinae , vol. 44 (2003), pp. 315–327.Google Scholar
Rothberger, F., Eine Äquivalenz zwischen der Kontinuumhypothese und der Existenz der Lusinschen und Sierpińskischen Mengen . Fundamenta Mathematicae , vol. 30 (1938), pp. 215–217.CrossRefGoogle Scholar
Shelah, S., Judah, H., and Miller, A., Sacks forcing, Laver forcing and Martin’s axiom . Archive for Mathematical Logic , vol. 31 (1992), pp. 145–161.Google Scholar
Shelah, S., Spinas, O., Goldstern, M., and Repický, M., On tree ideals . Proceedings of the American Mathematical Society , vol. 123 (1995), pp. 1573–1581.Google Scholar
Jech, T., Set Theory , millennium ed, Springer-Verlag, Berlin, 2003.Google Scholar
Weiss, T. and Kysiak, M., Small subsets of the reals and tree forcing notions . Proceedings of the American Mathematical Society , vol. 132 (2003), pp. 251–259.Google Scholar
Wohofsky, W., Brendle, J., and Khomskii, Y., Cofinalities of Marczewski-like ideals . Colloquium Mathematicum , vol. 150 (2017), pp. 1–10.Google Scholar