Strong Fubini properties of ideals

Ireneusz Recław¹ ; Piotr Zakrzewski¹

¹

Fundamenta Mathematicae, Tome 159 (1999) no. 2, pp. 135-152.

Résumé

 Let I and J be σ-ideals on Polish spaces X and Y, respectively. We say that the pair 〈I,J〉 has the Strong Fubini Property (SFP) if for every set D ⊆ X× Y with measurable sections, if all its sections $D_x = {y: 〈x,y〉 ∈ D}$ are in J, then the sections $D^y = {x: 〈x,y〉 ∈ D}$ are in I for every y outside a set from J (``measurable" means being a member of the σ-algebra of Borel sets modulo sets from the respective σ-ideal). We study the question of which pairs of σ-ideals have the Strong Fubini Property. Since CH excludes this phenomenon completely, sufficient conditions for SFP are always independent of ZFC.  We show, in particular, that:  • if there exists a Lusin set of cardinality the continuum and every set of reals of cardinality the continuum contains a one-to-one Borel image of a non-meager set, then 〈MGR(X), J〉 has SFP for every J generated by a hereditary $п^1_1$ (in the Effros Borel structure) family of closed subsets of Y (MGR(X) is the σ-ideal of all meager subsets of X),  • if there exists a Sierpiński set of cardinality the continuum and every set of reals of cardinality the continuum contains a one-to-one Borel image of a set of positive outer Lebesgue measure, then $〈NULL_μ, J〉$ has SFP if either $J= NULL_ν$ or J is generated by any of the following families of closed subsets of Y ($NULL_μ$ is the σ-ideal of all subsets of X having outer measure zero with respect to a Borel σ-finite continuous measure μ on X):  (i) all compact sets,  (ii) all closed sets in $NULL_ν$ for a Borel σ-finite continuous measure ν on Y,  (iii) all closed subsets of a $п^1_1$ set A ⊆ Y.

Détail

DOI : 10.4064/fm-159-2-135-152

Keywords: Polish space, Strong Fubini Property, σ-ideal, cardinal coefficients, measurability