On microscopic sets and Fubini Property in all directions

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\mathbb{R}^{2} $ with the following property: $E_{x}\in\mathcal{M} $ for any $x\in\mathbb{R} $ and $\{y;E^{y}\notin\mathcal{N}\}\notin\mathcal{M} $, for vertical sections $E_{x}=\{y;(x,y)\in E\} $ and horizontal sections $E^{y}=\{x;(x,y)\in E\} $ for $E\subset\mathbb{R}^{2} $. Thus $(\mathcal{N},\mathcal{M}) $ does not satisfy Fubini Property. In this paper we obtain such Borel set E, that $\{y;E^{y}\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.

On some generic small Cantor spaces

“…Microscopic subsets of the real line were introduced in [1]. Since then, they have been widely studied (see, for instance, [2], [3], [13], [12], [14], [18], [15], [16], [17], [19], [11], [21], [22]). The collection of microscopic subsets of the real line is a σ-ideal and it is a proper subset of the family of the Hausdorff zero dimensional sets ( [2], [14]).…”

“…In [3], the authors consider the family CS of symmetric Cantor subsets of [0, 1], and among other results, they obtain properties concerning the subfamily of microscopic sets. In [13], [12] and [15], the authors generalise the notion of a microscopic set in R. In [21] and [22] some Fubini type properties involving microscopic fibers are studied. In [18] and in [19], the notion of a microscopic set in the plane is investigated.…”