Haar null and non-dominating sets

Sławomir Solecki

Displaying similar documents to “Haar null and non-dominating sets”

$C (X)$ can sometimes determine $X$ without $X$ being realcompact

Melvin Henriksen, Biswajit Mitra (2005)

Commentationes Mathematicae Universitatis Carolinae

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As usual $C (X)$ will denote the ring of real-valued continuous functions on a Tychonoff space $X$ . It is well-known that if $X$ and $Y$ are realcompact spaces such that $C (X)$ and $C (Y)$ are isomorphic, then $X$ and $Y$ are homeomorphic; that is $C (X)$ $X$ . The restriction to realcompact spaces stems from the fact that $C (X)$ and $C (υ X)$ are isomorphic, where $υ X$ is the (Hewitt) realcompactification of $X$ . In this note, a class of locally compact spaces $X$ that includes properly the class of locally...

On nonmeasurable images

Robert Rałowski, Szymon Żeberski (2010)

Czechoslovak Mathematical Journal

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Let $(X, 𝕀)$ be a Polish ideal space and let $T$ be any set. We show that under some conditions on a relation $R \subseteq T^{2} \times X$ it is possible to find a set $A \subseteq T$ such that $R (A^{2})$ is completely $𝕀$ -nonmeasurable, i.e, it is $𝕀$ -nonmeasurable in every positive Borel set. We also obtain such a set $A \subseteq T$ simultaneously for continuum many relations ${(R_{α})}_{α < 2^{ω}} .$ Our results generalize those from the papers of K. Ciesielski, H. Fejzić, C. Freiling and M. Kysiak.

A characterization of the meager ideal

Piotr Zakrzewski (2015)

Commentationes Mathematicae Universitatis Carolinae

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We give a classical proof of the theorem stating that the $σ$ -ideal of meager sets is the unique $σ$ -ideal on a Polish group, generated by closed sets which is invariant under translations and ergodic.

Cardinal characteristics of the ideal of Haar null sets

Taras O. Banakh (2004)

Commentationes Mathematicae Universitatis Carolinae

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We calculate the cardinal characteristics of the $σ$ -ideal $ℋ 𝒩 (G)$ of Haar null subsets of a Polish non-locally compact group $G$ with invariant metric and show that $cov (ℋ 𝒩 (G)) \leq 𝔟 \leq max {𝔡, non (𝒩)} \leq non (ℋ 𝒩 (G)) \leq cof (ℋ 𝒩 (G)) > min {𝔡, non (𝒩)}$ . If $G = \prod_{n \geq 0} G_{n}$ is the product of abelian locally compact groups $G_{n}$ , then $add (ℋ 𝒩 (G)) = add (𝒩)$ , $cov (ℋ 𝒩 (G)) = min {𝔟, cov (𝒩)}$ , $non (ℋ 𝒩 (G)) = max {𝔡, non (𝒩)}$ and $cof (ℋ 𝒩 (G)) \geq cof (𝒩)$ , where $𝒩$ is the ideal of Lebesgue null subsets on the real line. Martin Axiom implies that $cof (ℋ 𝒩 (G)) > 2^{ℵ_{0}}$ 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...

Normal restrictions of the noncofinal ideal on $P_{κ} (λ)$

Pierre Matet (2013)

Fundamenta Mathematicae

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We discuss the problem of whether there exists a restriction of the noncofinal ideal on $P_{κ} (λ)$ that is normal.

Non-locally compact Polish groups and two-sided translates of open sets

Maciej Malicki (2008)

Fundamenta Mathematicae

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This paper is devoted to the following question. Suppose that a Polish group G has the property that some non-empty open subset U is covered by finitely many two-sided translates of every other non-empty open subset of G. Is then G necessarily locally compact? Polish groups which do not have the above property are called strongly non-locally compact. We characterize strongly non-locally compact Polish subgroups of $S_{\infty}$ in terms of group actions, and prove that certain natural classes of...

Borel sets with σ-compact sections for nonseparable spaces

Petr Holický (2008)

Fundamenta Mathematicae

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We prove that every (extended) Borel subset E of X × Y, where X is complete metric and Y is Polish, can be covered by countably many extended Borel sets with compact sections if the sections $E_{x} = y \in Y : (x, y) \in E$ , x ∈ X, are σ-compact. This is a nonseparable version of a theorem of Saint Raymond. As a by-product, we get a proof of Saint Raymond’s result which does not use transfinite induction.

An ideal characterization of when a subspace of certain Banach spaces has the metric compact approximation property

J. Cabello, E. Nieto (1998)

Studia Mathematica

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C.-M. Cho and W. B. Johnson showed that if a subspace E of $ℓ_{p}$ , 1 < p < ∞, has the compact approximation property, then K(E) is an M-ideal in ℒ(E). We prove that for every r,s ∈ ]0,1] with $r^{2} + s^{2} < 1$ , the James space can be provided with an equivalent norm such that an arbitrary subspace E has the metric compact approximation property iff there is a norm one projection P on ℒ(E)* with Ker P = K(E)⊥ satisfying ∥⨍∥ ≥ r∥Pf∥ + s∥φ - Pf∥ ∀⨍ ∈ ℒ(E)*. A similar result is proved for subspaces of...

Spectrum of commutative Banach algebras and isomorphism of C*-algebras related to locally compact groups

Zhiguo Hu (1998)

Studia Mathematica

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Let A be a semisimple commutative regular tauberian Banach algebra with spectrum $Σ_{A}$ . In this paper, we study the norm spectra of elements of $\bar{s p a n} Σ_{A}$ and present some applications. In particular, we characterize the discreteness of $Σ_{A}$ in terms of norm spectra. The algebra A is said to have property (S) if, for all $φ \in \bar{} Σ_{A} 0$ , φ has a nonempty norm spectrum. For a locally compact group G, let $ℳ_{2}^{d} (Ĝ)$ denote the C*-algebra generated by left translation operators on $L^{2} (G)$ and $G_{d}$ denote the discrete group G. We prove...