Fubini Property for microscopic sets

Abstract

In 2000 I. Recaw and P. Zakrzewski introduced the notion of Fubini
Property for the pair ($\mathcal{I},\mathcal{J}$) of two $\sigma$-ideals in a following way. Let $\mathcal{I}$
and $\mathcal{J}$ be two $\sigma$-ideals on Polish spaces $X$ and $Y$ , respectively. The
pair ($\mathcal{I},\mathcal{J}$) has the Fubini Property (FP) if for every Borel subset $B$
of $ X  \times Y$ , if all its vertical sections

$Bx ={y \in  Y : (x; y) \in B}$ are in $\mathcal{J}$, then the set of all $y \in Y$ for which horizontal section $B^y = {x \in X : (x; y) \in B}$

does not belong to $\mathcal{I}$, is a set from $\mathcal{J}$, i.e.
$${y \in Y : B^y \not\in \mathcal{I} \in \mathcal{J}.$$

The Fubini property for the $\sigma$-ideal $\mathcal{M} of microscopic sets is considered and the proof that the pair $(M;M)$ does not satisfy (FP) is
given.