Marczewski sets, measure and the Baire property
نویسندگان
چکیده
منابع مشابه
Ramsey, Lebesgue, and Marczewski Sets and the Baire Property
We investigate the completely Ramsey, Lebesgue, and Marczewski σ-algebras and their relations to the Baire property in the Ellentuck and density topologies. Two theorems concerning the Marczewski σ-algebra (s) are presented. Theorem. In the density topology D, (s) coincides with the σ-algebra of Lebesgue measurable sets. Theorem. In the Ellentuck topology on [ω] , (s)0 is a proper subset of the...
متن کاملExtending Baire property by uncountably many sets
We prove that if ZFC is consistent so is ZFC + “for any sequence (An) of subsets of a Polish space 〈X, τ〉 there exists a separable metrizable topology τ ′ on X with B(X, τ) ⊆ B(X, τ ′), MGR(X, τ ′) ∩ B(X, τ) = MGR(X, τ) ∩B(X, τ) and An Borel in τ ′ for all n.” This is a category analogue of a theorem of Carlson on the possibility of extending Lebesgue measure to any countable collection of sets...
متن کاملOn Marczewski Sets
Using the methods of Brown and Walsh, we get condition guaranteeing that, for an ideal I of sets in a perfect Polish space some (s 0) sets are not in I. A few examples and corollaries are given. 0. Introduction Papers Br], W1], W2] and C] made a signiicant progress in the studying of (s 0) sets introduced by Marczewski in Sz]. One of the main results states that there exists a nonmeasurable (s ...
متن کاملSolutions to Congruences Using Sets with the Property of Baire
Hausdorff’s paradoxical decomposition of a sphere with countably many points removed (the main precursor of the Banach-Tarski paradox) actually produced a partition of this set into three pieces A, B, C such that A is congruent to B (i.e., there is an isometry of the set which sends A to B), B is congruent to C, and A is congruent to B ∪ C. While refining the Banach-Tarski paradox, R. Robinson ...
متن کاملBanach-tarski Decompositions Using Sets with the Property of Baire
Perhaps the most strikingly counterintuitive theorem in mathematics is the "Banach-Tarski paradox": A ball in R3 can be decomposed into finitely many pieces which can be rearranged by rigid motions and reassembled to form two balls of the same size as the original. The Axiom of Choice is used to construct the decomposition; the "paradox" is resolved by noting that the pieces cannot be Lebesgue ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Fundamenta Mathematicae
سال: 1988
ISSN: 0016-2736,1730-6329
DOI: 10.4064/fm-129-2-83-89