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Generalized projections of Borel and analytic sets

100%

Balcerzak M.

Colloquium Mathematicum

1996

tom 71

nr 1

47-53

For a σ-ideal I of sets in a Polish space X and for A ⊆ $X^2$, we consider the generalized projection 𝛷(A) of A given by 𝛷(A) = {x ∈ X: A_x ∉ I}, where $A_x$ ={y ∈ X: 〈x,y〉∈ A}. We study the behaviour of 𝛷 with respect to Borel and analytic sets in the case when I is a $∑_{2}^{0}$-supported σ-ideal. In particular, we give an alternative proof of the recent result of Kechris showing that 𝛷 [$∑_{1}^{1}(X^2)]=∑_{1}^{1}(X)$ for a wide class of $∑_{2}^{0}$-supported σ-ideals.

Strong Fubini properties of ideals

88%

Recław I., Zakrzewski P.

Fundamenta Mathematicae

1999

tom 159

nr 2

135-152

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.

Ograniczanie wyników

1 Colloquium Mathematicum

1 Fundamenta Mathematicae

1 Balcerzak M.

1 Recław I.

1 Zakrzewski P.

1 1999

1 1996

Wyniki wyszukiwania

Generalized projections of Borel and analytic sets

Strong Fubini properties of ideals