Fubini Property for Microscopic Sets

In 2000, I. Recław and P. Zakrzewski introduced the notion of Fubini Property for the pair (I,J) of two σ-ideals in the following way. Let I and J be two σ-ideals on Polish spaces X and Y, respectively. The pair (I,J) has the Fubini Property (FP) if for every Borel subset B of X×Y such that all its vertical sections B_x = {y ∈ Y : (x, y) ∈ B} are in J, then the set of all y ∈ Y, for which horizontal section By = {x ∈ X : (x, y) ∈ B} does not belong to I, is a set from J, i.e.,

{y ∈ Y : B^y ∉ I} ∈ J.

The Fubini property for the σ-ideal M of microscopic sets is considered and the proof that the pair (M,M) does not satisfy (FP) is given.