Uncountable Versions of AC and Lebesgue Nonmeasurable Sets

2014 ◽  
pp. 57-74
Keyword(s):  
Nonmeasurable Sets

scholarly journals Paradoxical characterization of Lebesgue nonmeasurable sets

Heliyon ◽  
2020 ◽  
Vol 6 (8) ◽  
pp. e04652
Author(s):  
Łukasz Kruk
Keyword(s):  
Nonmeasurable Sets

scholarly journals Bernstein sets with algebraic properties

2009 ◽  
Vol 7 (4) ◽  
Author(s):  
Marcin Kysiak
Keyword(s):  
Algebraic Properties ◽  
Bernstein Set ◽  
Nonmeasurable Sets

AbstractWe construct Bernstein sets in ℝ having some additional algebraic properties. In particular, solving a problem of Kraszewski, Rałowski, Szczepaniak and Żeberski, we construct a Bernstein set which is a < c-covering and improve some other results of Rałowski, Szczepaniak and Żeberski on nonmeasurable sets.

scholarly journals NONMEASURABLE SETS AND UNIONS WITH RESPECT TO TREE IDEALS

2020 ◽  
Vol 26 (1) ◽  
pp. 1-14
Author(s):  
MARCIN MICHALSKI ◽  
ROBERT RAŁOWSKI ◽  
SZYMON ŻEBERSKI
Keyword(s):  
Real Line ◽  
Regular Cardinal ◽  
Baire Space ◽  
The Real ◽  
Nonmeasurable Sets

AbstractIn 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 $cl_0$ .

scholarly journals Exceptional directions for Sierpiński's nonmeasurable sets

1992 ◽  
Vol 140 (3) ◽  
pp. 237-245
Author(s):  
B. Kirchheim ◽  
Tomasz Natkaniec
Keyword(s):  
Nonmeasurable Sets
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