## Real Analysis Exchange

### On Monotone Presentations of Borel Sets

#### Abstract

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

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

Dates
First available in Project Euclid: 29 October 2009

http://projecteuclid.org/euclid.rae/1256835189

Mathematical Reviews number (MathSciNet)
MR2569189

Zentralblatt MATH identifier
1187.03038

#### Citation

Mátrai, Tamás; Zelený, Miroslav. On Monotone Presentations of Borel Sets. Real Analysis Exchange 34 (2008), no. 2, 311--318. http://projecteuclid.org/euclid.rae/1256835189.

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