In 2000 I. Recaw and P. Zakrzewski introduced the notion of FubiniProperty 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. Thepair ($\mathcal{I},\mathcal{J}$) 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) isgiven.
@article{398,
title = {Fubini Property for microscopic sets},
journal = {Tatra Mountains Mathematical Publications},
volume = {65},
year = {2016},
doi = {10.2478/tatra.v65i0.398},
language = {EN},
url = {http://dml.mathdoc.fr/item/398}
}
Paszkiewicz, Adam; Wagner-Bojakowska, Elżbieta. Fubini Property for microscopic sets. Tatra Mountains Mathematical Publications, Tome 65 (2016) . doi : 10.2478/tatra.v65i0.398. http://gdmltest.u-ga.fr/item/398/