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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 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 ) ) ≤ 𝔟 ≤ max { 𝔡 , non ( 𝒩 ) } ≤ non ( ℋ 𝒩 ( G ) ) ≤ cof ( ℋ 𝒩 ( G ) ) > min { 𝔡 , non ( 𝒩 ) } . If G = ∏ n ≥ 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 ) ) ≥ 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 question of...

Covering locally compact groups by less than 2 ω many translates of a compact nullset

Márton Elekes, Árpád Tóth (2007)

Fundamenta Mathematicae

Gruenhage asked if it was possible to cover the real line by less than continuum many translates of a compact nullset. Under the Continuum Hypothesis the answer is obviously negative. Elekes and Stepr mans gave an affirmative answer by showing that if C E K is the well known compact nullset considered first by Erdős and Kakutani then ℝ can be covered by cof() many translates of C E K . As this set has no analogue in more general groups, it was asked by Elekes and Stepr mans whether such a result holds for...

Ergodicité et pureté des produits de Riesz

François Parreau (1990)

Annales de l'institut Fourier

On montre que les produits de Riesz sur le tore sont des mesures ergodiques sous une condition de lacunarité pour les fréquences, indépendamment de toute propriété arithmétique, et que cette condition est la meilleure possible de ce point de vue. On établit un critère analogue pour la propriété de pureté discutés précédemment par B. Host et l’auteur, ce qui fournit l’exemple d’une mesure pure étrangère à toutes ses translatées et en particulier non ergodique.

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