Fubini Property for Microscopic Sets

Abstract

ABSTRACT. In 2000, I. Rec law 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 B y = 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.

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.…”