Real Analysis Exchange

Absolute Null Subsets of the Plane with Bad Orthogonal Projections

Alexander Kharazishvili

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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

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

First available in Project Euclid: 29 March 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Absolute null set Bernstein set generalized Luzin set Hamel basis


Kharazishvili, Alexander. Absolute Null Subsets of the Plane with Bad Orthogonal Projections. Real Anal. Exchange 41 (2016), no. 1, 233--244.

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