Strong Fubini properties of ideals

 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 ${\displaystyle D_x=\{y:⟨x,y⟩\in D\}}$ are in J, then the sections ${\displaystyle D^y=\{x:⟨x,y⟩\in D\}}$ 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 ${\displaystyle {п}^1\_1}$ (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 ${\displaystyle ⟨NUL{L}_{\mu },J⟩}$ has SFP if either ${\displaystyle J=NUL{L}_{\nu }}$ or J is generated by any of the following families of closed subsets of Y (${\displaystyle NUL{L}_{\mu }}$ 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 ${\displaystyle NUL{L}_{\nu }}$ for a Borel σ-finite continuous measure ν on Y,  (iii) all closed subsets of a ${\displaystyle {п}^1\_1}$ set A ⊆ Y.