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Cardinal characteristics of the ideal of Haar null sets

Taras O. Banakh (2004)

Commentationes Mathematicae Universitatis Carolinae

We calculate the cardinal characteristics of the σ -ideal 𝒩 ( G ) of Haar null subsets of a Polish non-locally compact group G with invariant metric and show that cov ( 𝒩 ( G ) ) 𝔟 max { 𝔡 , non ( 𝒩 ) } non ( 𝒩 ( G ) ) cof ( 𝒩 ( G ) ) > min { 𝔡 , non ( 𝒩 ) } . If G = n 0 G n is the product of abelian locally compact groups G n , then add ( 𝒩 ( G ) ) = add ( 𝒩 ) , cov ( 𝒩 ( G ) ) = min { 𝔟 , cov ( 𝒩 ) } , non ( 𝒩 ( G ) ) = max { 𝔡 , non ( 𝒩 ) } and cof ( 𝒩 ( G ) ) cof ( 𝒩 ) , where 𝒩 is the ideal of Lebesgue null subsets on the real line. Martin Axiom implies that cof ( 𝒩 ( G ) ) > 2 0 and hence G contains a Haar null subset that cannot be enlarged to a Borel or projective Haar null subset of G . This gives a negative (consistent) answer to a question of...

Convexity and almost convexity in groups

Witold Jarczyk (2013)

Banach Center Publications

We give a review of results proved and published mostly in recent years, concerning real-valued convex functions as well as almost convex functions defined on a (not necessarily convex) subset of a group. Analogues of such classical results as the theorems of Jensen, Bernstein-Doetsch, Blumberg-Sierpiński, Ostrowski, and Mehdi are presented. A version of the Hahn-Banach theorem with a convex control function is proved, too. We also study some questions specific for the group setting, for instance...

Embedding a topological group into its WAP-compactification

Stefano Ferri, Jorge Galindo (2009)

Studia Mathematica

We prove that the topology of the additive group of the Banach space c₀ is not induced by weakly almost periodic functions or, what is the same, that this group cannot be represented as a group of isometries of a reflexive Banach space. We show, in contrast, that additive groups of Schwartz locally convex spaces are always representable as groups of isometries on some reflexive Banach space.

On a generalization of Abelian sequential groups

Saak S. Gabriyelyan (2013)

Fundamenta Mathematicae

Let (G,τ) be a Hausdorff Abelian topological group. It is called an s-group (resp. a bs-group) if there is a set S of sequences in G such that τ is the finest Hausdorff (resp. precompact) group topology on G in which every sequence of S converges to zero. Characterizations of Abelian s- and bs-groups are given. If (G,τ) is a maximally almost periodic (MAP) Abelian s-group, then its Pontryagin dual group ( G , τ ) is a dense -closed subgroup of the compact group ( G d ) , where G d is the group G with the discrete...

On characterized subgroups of Abelian topological groups X and the group of all X -valued null sequences

S. S. Gabriyelyan (2014)

Commentationes Mathematicae Universitatis Carolinae

Let X be an Abelian topological group. A subgroup H of X is characterized if there is a sequence 𝐮 = { u n } in the dual group of X such that H = { x X : ( u n , x ) 1 } . We reduce the study of characterized subgroups of X to the study of characterized subgroups of compact metrizable Abelian groups. Let c 0 ( X ) be the group of all X -valued null sequences and 𝔲 0 be the uniform topology on c 0 ( X ) . If X is compact we prove that c 0 ( X ) is a characterized subgroup of X if and only if X 𝕋 n × F , where n 0 and F is a finite Abelian group. For every compact Abelian...

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