Real Analysis Exchange

Absolute Null Subsets of the Plane with Bad Orthogonal Projections

Alexander Kharazishvili

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Abstract

Under Martin’s Axiom, it is proved that there exists an absolute null subset of the Euclidean plane $\mathbb{R}^2$, the orthogonal projections of which on all straight lines in $\mathbb{R}^2$ are absolutely nonmeasurable. A similar but weaker result holds true within the framework of ZFC set theory.

Article information

Source
Real Anal. Exchange, Volume 41, Number 1 (2016), 233-244.

Dates
First available in Project Euclid: 29 March 2017

Permanent link to this document
https://projecteuclid.org/euclid.rae/1490752825

Mathematical Reviews number (MathSciNet)
MR3511944

Zentralblatt MATH identifier
1384.28003

Subjects
Primary: 28A05: Classes of sets (Borel fields, $\sigma$-rings, etc.), measurable sets, Suslin sets, analytic sets [See also 03E15, 26A21, 54H05] 28D05: Measure-preserving transformations
Secondary: 03E25: Axiom of choice and related propositions

Keywords
Absolute null set Bernstein set generalized Luzin set Hamel basis

Citation

Kharazishvili, Alexander. Absolute Null Subsets of the Plane with Bad Orthogonal Projections. Real Anal. Exchange 41 (2016), no. 1, 233--244. https://projecteuclid.org/euclid.rae/1490752825


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