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Strong Fubini properties of ideals

Ireneusz Recław; Piotr Zakrzewski

Fundamenta Mathematicae (1999)

  • Volume: 159, Issue: 2, page 135-152
  • ISSN: 0016-2736

Abstract

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 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 ⟨ N U L L μ , J ⟩ has SFP if either J = N U L L ν or J is generated by any of the following families of closed subsets of Y ( N U L L 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 N U L L ν for a Borel σ-finite continuous measure ν on Y,  (iii) all closed subsets of a п 1 1 set A ⊆ Y.

How to cite

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Recław, Ireneusz, and Zakrzewski, Piotr. "Strong Fubini properties of ideals." Fundamenta Mathematicae 159.2 (1999): 135-152. <http://eudml.org/doc/212325>.

@article{Recław1999,
abstract = { 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.},
author = {Recław, Ireneusz, Zakrzewski, Piotr},
journal = {Fundamenta Mathematicae},
keywords = {Polish space; Strong Fubini Property; σ-ideal; cardinal coefficients; measurability; -ideals; strong Fubini property; meager sets; null sets; Polish spaces; Borel sets; Lusin set; Sierpiński set},
language = {eng},
number = {2},
pages = {135-152},
title = {Strong Fubini properties of ideals},
url = {http://eudml.org/doc/212325},
volume = {159},
year = {1999},
}

TY - JOUR
AU - Recław, Ireneusz
AU - Zakrzewski, Piotr
TI - Strong Fubini properties of ideals
JO - Fundamenta Mathematicae
PY - 1999
VL - 159
IS - 2
SP - 135
EP - 152
AB -  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.
LA - eng
KW - Polish space; Strong Fubini Property; σ-ideal; cardinal coefficients; measurability; -ideals; strong Fubini property; meager sets; null sets; Polish spaces; Borel sets; Lusin set; Sierpiński set
UR - http://eudml.org/doc/212325
ER -

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