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# Mathematica Slovaca

Editor-in-Chief: Pulmannová, Sylvia

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Volume 68, Issue 5

# On microscopic sets and Fubini Property in all directions

Published Online: 2018-10-20 | DOI: https://doi.org/10.1515/ms-2017-0165

## Abstract

For the σ-ideal $\mathcal{N}$ of nullsets and σ-ideal $\mathcal{M}$ of microscopic sets, it was recently obtained that there exists a Borel set $E\subset {ℝ}^{2}$ with the following property: ${E}_{x}\in \mathcal{M}$ for any $x\in ℝ$ and $\left\{y;{E}^{y}\notin \mathcal{N}\right\}\notin \mathcal{M}$, for vertical sections ${E}_{x}=\left\{y;\left(x,y\right)\in E\right\}$ and horizontal sections ${E}^{y}=\left\{x;\left(x,y\right)\in E\right\}$ for $E\subset {ℝ}^{2}$. Thus $\left(\mathcal{N},\mathcal{M}\right)$ does not satisfy Fubini Property. In this paper we obtain such Borel set E, that $\left\{y;{E}^{y}\notin \mathcal{N}\right\}\notin \mathcal{M}$ and all non-horizontal (in a natural sense) sections of E are in $\mathcal{M}$. Other Fubini type properties, with conditions written for all directions are also discussed.

MSC 2010: Primary 28A05; Secondary 03E15; 26A30

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Accepted: 2017-07-07

Published Online: 2018-10-20

Published in Print: 2018-10-25

Communicated by David Buhagiar

Citation Information: Mathematica Slovaca, Volume 68, Issue 5, Pages 1041–1048, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918,

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