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## Fundamenta Mathematicae

1999 | 159 | 2 | 135-152
Tytuł artykułu

### Strong Fubini properties of ideals

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Treść / Zawartość
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Języki publikacji
EN
Abstrakty
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.
Słowa kluczowe
EN
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
135-152
Opis fizyczny
Daty
wydano
1999
otrzymano
1998-04-01
Twórcy
autor
autor
• Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland, piotrzak@mimuw.edu.pl
Bibliografia
• [1] T. Bartoszyński and H. Judah, Set Theory. On the Structure of the Real Line, A K Peters, 1995.
• [2] R. H. Bing, W. W. Bledsoe and R. D. Mauldin, Sets generated by rectangles, Pacific J. Math. 51 (1974), 27-36.
• [3] J. Brzuchowski, J. Cichoń and B. Węglorz, Some applications of strong Lusin sets, Compositio Math. 43 (1981), 217-224.
• [4] T. Carlson, Extending Lebesgue measure by infinitely many sets, Pacific J. Math. 115 (1984), 33-45.
• [5] K. Eda, M. Kada and Y. Yuasa, The tightness about sequential fans and combinatorial properties, J. Math. Soc. Japan 49 (1997), 181-187.
• [6] C. Freiling, Axioms of symmetry: throwing the darts at the real line, J. Symbolic Logic 51 (1986), 190-220.
• [7] D. H. Fremlin, Measure-additive coverings and measurable selectors, Dissertationes Math. 260 (1987).
• [8] D. H. Fremlin, Real-valued-measurable cardinals, in: Set Theory of the Reals, H. Judah (ed.), Israel Math. Conf. Proc. 6 (1993), 151-304.
• [9] H. Friedman, A consistent Fubini-Tonelli theorem for nonmeasurable functions, Illinois J. Math. 24 (1980), 390-395.
• [10] P. R. Halmos, Measure Theory, Van Nostrand, 1950.
• [11] M. Kada and Y. Yuasa, Cardinal invariants about shrinkability of unbounded sets, Topology Appl. 74 (1996), 215-223.
• [12] A. Kamburelis, A new proof of the Gitik-Shelah theorem, Israel J. Math. 72 (1990), 373-380.
• [13] A. Kanamori and M. Magidor, The evolution of large cardinal axioms in set theory, in: Higher Set Theory, Lecture Notes in Math. 669, Springer, 1978, 99-275.
• [14] A. S. Kechris, Classical Descriptive Set Theory, Grad. Texts in Math. 156, Springer, 1995.
• [15] A. W. Miller, Mapping a set of reals onto the reals, J. Symbolic Logic 48 (1983), 575-584.
• [16] I. Recław and P. Zakrzewski, Fubini properties of ideals, submitted for publication.
• [17] I. Recław and P. Zakrzewski, Strong Fubini properties of ideals, preprint P 97-10, Institute of Math., Warsaw University.
• [18] J. Shipman, Cardinal conditions for strong Fubini theorems, Trans. Amer. Math. Soc. 321 (1990), 465-481.
• [19] P. Zakrzewski, Strong Fubini axioms from measure extension axioms, Comment. Math. Univ. Carolin. 33 (1992), 291-297.
• [20] P. Zakrzewski, Extending Baire Property by countably many sets, submitted for publication.
• [21] P. Zakrzewski, Fubini properties of ideals and forcing, to appear.
Typ dokumentu
Bibliografia
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