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    Wyświetlanie 1-2 z 2
    Tytuł:
    Strong Fubini properties of ideals
    Autorzy:
    Recław, Ireneusz
    Zakrzewski, Piotr
    Powiązania:
    https://bibliotekanauki.pl/articles/1205274.pdf
    Data publikacji:
    1999
    Wydawca:
    Polska Akademia Nauk. Instytut Matematyczny PAN
    Tematy:
    Polish space
    Strong Fubini Property
    σ-ideal
    cardinal coefficients
    measurability
    Opis:

     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.
    Źródło:
    Fundamenta Mathematicae; 1999, 159, 2; 135-152
    0016-2736
    Pojawia się w:
    Fundamenta Mathematicae
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Convexity ranks in higher dimensions
    Autorzy:
    Kojman, Menachem
    Powiązania:
    https://bibliotekanauki.pl/articles/1205064.pdf
    Data publikacji:
    2000
    Wydawca:
    Polska Akademia Nauk. Instytut Matematyczny PAN
    Tematy:
    convexity
    convexity number
    Polish vector space
    continuum hypothesis
    Cantor-Bendixson degree
    Opis:
    A subset of a vector space is called countably convex if it is a countable union of convex sets. Classification of countably convex subsets of topological vector spaces is addressed in this paper. An ordinal-valued rank function ϱ is introduced to measure the complexity of local nonconvexity points in subsets of topological vector spaces. Then ϱ is used to give a necessary and sufficient condition for countable convexity of closed sets. Theorem. Suppose that S is a closed subset of a Polish linear space. Then S is countably convex if and only if there exists $α < ω_1$ so that ϱ(x) < α for all x ∈ S. Classification of countably convex closed subsets of Polish linear spaces follows then easily. A similar classification (by a different rank function) was previously known for closed subset of $ℝ^2$ [3]. As an application of ϱ to Banach space geometry, it is proved that for every $α < ω_1$, the unit sphere of C(ωα) with the sup-norm has rank α. Furthermore, a countable compact metric space K is determined by the rank of the unit sphere of C(K) with the natural sup-norm: Theorem. If $K_1,K_1$ are countable compact metric spaces and $S_i$ is the unit sphere in $C(K_i)$ with the sup-norm, i = 1,2, then $ϱ(S_1) = ϱ(S_2)$ if and only if $K_1$ and $K_2$ are homeomorphic. Uncountably convex closed sets are also studied in dimension n > 2 and are seen to be drastically more complicated than uncountably convex closed subsets of $ℝ^2$
    Źródło:
    Fundamenta Mathematicae; 2000, 164, 2; 143-163
    0016-2736
    Pojawia się w:
    Fundamenta Mathematicae
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
      Wyświetlanie 1-2 z 2

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