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Inserting measurable functions precisely

Javier Gutiérrez García, Tomasz Kubiak (2014)

Czechoslovak Mathematical Journal

A family of subsets of a set is called a σ -topology if it is closed under arbitrary countable unions and arbitrary finite intersections. A σ -topology is perfect if any its member (open set) is a countable union of complements of open sets. In this paper perfect σ -topologies are characterized in terms of inserting lower and upper measurable functions. This improves upon and extends a similar result concerning perfect topologies. Combining this characterization with a σ -topological version of Katětov-Tong...

Kadec norms and Borel sets in a Banach space

M. Raja (1999)

Studia Mathematica

We introduce a property for a couple of topologies that allows us to give simple proofs of some classic results about Borel sets in Banach spaces by Edgar, Schachermayer and Talagrand as well as some new results. We characterize the existence of Kadec type renormings in the spirit of the new results for LUR spaces by Moltó, Orihuela and Troyanski.

La genèse du théorème de recouvrement de Borel

Bernard Maurey, Jean-Pierre Tacchi (2005)

Revue d'histoire des mathématiques

Nous nous proposons de rendre à Émile Borel le mérite d’avoir considéré le premier un recouvrement d’un segment de droite par une suite infinie d’intervalles et prouvé que l’on peut en extraire un sous-recouvrement fini. L’appellation de théorème de Heine-Borel souvent donnée à ce résultat, en référence à un article de Heine de 1872, conduit à sous-estimer les différences avec le théorème sur la continuité uniforme (dont une première version peut être attribuée à Dirichlet, en 1854) ; cette dénomination...

Less than 2 ω many translates of a compact nullset may cover the real line

Márton Elekes, Juris Steprāns (2004)

Fundamenta Mathematicae

We answer a question of Darji and Keleti by proving that there exists a compact set C₀ ⊂ ℝ of measure zero such that for every perfect set P ⊂ ℝ there exists x ∈ ℝ such that (C₀+x) ∩ P is uncountable. Using this C₀ we answer a question of Gruenhage by showing that it is consistent with ZFC (as it follows e.g. from c o f ( ) < 2 ω ) that less than 2 ω many translates of a compact set of measure zero can cover ℝ.

Marczewski-Burstin Representations of Boolean Algebras Isomorphic to a Power Set

Artur Bartoszewicz (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

The paper contains some sufficient conditions for Marczewski-Burstin representability of an algebra 𝓐 of sets which is isomorphic to 𝓟(X) for some X. We characterize those algebras of sets which are inner MB-representable and isomorphic to a power set. We consider connections between inner MB-representability and hull property of an algebra isomorphic to 𝓟 (X) and completeness of an associated quotient algebra. An example of an infinite universally MB-representable algebra is given.

Currently displaying 161 – 180 of 415