Real Analysis Exchange

On Monotone Presentations of Borel Sets

Tamás Mátrai and Miroslav Zelený

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If $A$ is a ${\bf \Sigma}^{0}_{\xi}$ set and $A_{n}$ $(n \omega)$ are Borel sets then we call $\{A_{n} \colon n \omega\}$ a presentation of $A$ if $A = \bigcup_{n \omega}A_{n}$ and $A_{n}$ $(n \omega)$ have lower Borel class than $A$ has. We show that for $2 \leq \xi \omega_{1}$ it is not possible to assign a presentation to ${\bf \Sigma}^{0}_{\xi}$ sets in a monotone way; i.e., it is not possible to define functions $f_{n} \colon {\bf \Sigma}^{0}_{\xi} \rightarrow {\bf \Pi}^{0}_{\xi}$ $(n \omega)$ such that for every $A \in {\bf \Sigma}^{0}_{\xi}$ we have $A = \bigcup_{n \omega}f_{n}(A)$ and $A, A' \in {\bf \Sigma}^{0}_{\xi}$, $A \subseteq A'$ implies $f_{n}(A) \subseteq f_{n}(A')$ $(n \omega)$. This answers a question of M\'arton Elekes in the negative. We also show the nonexistence of monotone presentation for Borel functions.

Article information

Real Anal. Exchange, Volume 34, Number 2 (2008), 311-318.

First available in Project Euclid: 29 October 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03E15: Descriptive set theory [See also 28A05, 54H05]
Secondary: 54H05: Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets) [See also 03E15, 26A21, 28A05] 28A05: Classes of sets (Borel fields, $\sigma$-rings, etc.), measurable sets, Suslin sets, analytic sets [See also 03E15, 26A21, 54H05] 28A10: Real- or complex-valued set functions 26A21: Classification of real functions; Baire classification of sets and functions [See also 03E15, 28A05, 54C50, 54H05] 54C50: Special sets defined by functions [See also 26A21]

Borel set Borel function canonical presentation monotone presentation


Mátrai, Tamás; Zelený, Miroslav. On Monotone Presentations of Borel Sets. Real Anal. Exchange 34 (2008), no. 2, 311--318.

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