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On some theorem on measurable selection

100%
Shragin I. V.
Commentationes Mathematicae
|
2007
|
tom 47
|
nr 2
EN
A new proof of some results of V.L. Levin on measurability of projection and on measurable selection is given.
2
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Completely nonmeasurable unions

100%
Rałowski R., Żeberski S.
Open Mathematics
|
2010
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tom 8
|
nr 4
683-687
EN
Assume that no cardinal κ < 2ω is quasi-measurable (κ is quasi-measurable if there exists a κ-additive ideal of subsets of κ such that the Boolean algebra P(κ)/ satisfies c.c.c.). We show that for a metrizable separable space X and a proper c.c.c. σ-ideal II of subsets of X that has a Borel base, each point-finite cover ⊆ $$ \mathbb{I} $$ of X contains uncountably many pairwise disjoint subfamilies , with $$ \mathbb{I} $$-Bernstein unions ∪ (a subset A ⊆ X is $$ \mathbb{I} $$-Bernstein if A and X \ A meet each Borel $$ \mathbb{I} $$-positive subset B ⊆ X). This result is a generalization of the Four Poles Theorem (see [1]) and results from [2] and [4].
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Strong Fubini properties of ideals

100%
Recław I., Zakrzewski P.
Fundamenta Mathematicae
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1999
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tom 159
|
nr 2
135-152
EN
 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.
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