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On microscopic sets and Fubini property in all directions

In: Mathematica Slovaca, vol. 68, no. 5
Adam Paszkiewicz
Detaily:
Rok, strany: 2018, 1041 - 1048
Kľúčové slová:
nullsets, microscopic sets, Fubini property, marginal distributions
O článku:
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 $E\subset\Bbb{R}2$ with the following property: $Ex\in\mathcal{M}$ for any $x\in\Bbb{R}$ and $\{y; Ey\notin\mathcal{N}\}\notin \mathcal{M}$, for vertical sections $Ex=\{y; (x, y)\in E\}$ and horizontal sections $Ey =\{x; (x, y)\in E\}$ for $E\subset\Bbb{R}2$. Thus $(\mathcal{N}, \mathcal{M})$ does not satisfy Fubini Property. In this paper we obtain such Borel set $E$, that $\{y; Ey\notin \mathcal{N}\}\notin \mathcal{M}$ 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.
Ako citovať:
ISO 690:
Paszkiewicz, A. 2018. On microscopic sets and Fubini property in all directions. In Mathematica Slovaca, vol. 68, no.5, pp. 1041-1048. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0165

APA:
Paszkiewicz, A. (2018). On microscopic sets and Fubini property in all directions. Mathematica Slovaca, 68(5), 1041-1048. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0165
O vydaní:
Publikované: 31. 10. 2018