Can ideals without ccc be interesting?

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.

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“…The same result can be obtained using [2], [7] and the prior properties of microscopic sets ([4, Proposition 20.14]).…”

Section: The Aim Of This Paper Is To Prove That the Pair (M M) Does supporting

confidence: 64%

Fubini Property for Microscopic Sets

Paszkiewicz

Wagner-Bojakowska

2016

Tatra Mountains Mathematical Publications

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“…By removing the requirement about Borel hulls from the above definition, we would obtain the notions of perfectly κ-small sets studied by Zakrzewski in [19] in the context of ccc σ-ideals. The particular case of Haar-1 sets has been investigated by Balcerzak in [1], where he introduced the so-called property (D) of an invariant σ-ideal I, which says that there is a σ-compact Haar-1 set not belonging to I. Moreover, Banakh, Lyaskovska and Repovš in [4] considered packing index of a set, which is closely connected to Haar-1 sets (namely, a Borel set is Haar-1 if and only if its packing index is uncountable).…”

Section: Clearlymentioning

confidence: 99%

Haar-smallest sets

Kwela

2019

Topology and its Applications

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“…There are other σ‐ideals with property (M): ideals on Polish spaces with a \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Sigma ^0_2$\end{document} base which are not ccc (see 12), some ideals defined by translations (see 2 and 13). For the further discussion on property (M) see 1, 2 and 4.…”

Section: Fubini Productsmentioning

confidence: 99%