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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 $cov (ℋ 𝒩 (G)) \leq 𝔟 \leq max {𝔡, non (𝒩)} \leq non (ℋ 𝒩 (G)) \leq cof (ℋ 𝒩 (G)) > min {𝔡, non (𝒩)}$ . 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 question of...

Central Idempotents in Measure Algebras.

Martin Moskowitz, Richard D. Mosak (1971)

Mathematische Zeitschrift

Christensen measurability of polynomial functions and convex functions of higher orders

Zbigniew Gajda (1986)

Annales Polonici Mathematici

Christensen measurable solutions of generalized Cauchy functional equations.

Zbigniew Gajda (1986)

Aequationes mathematicae

Christensen measurable solutions of generalized Cauchy functional equations. (Short Communications).

Zbigniew Gajda (1986)

Aequationes mathematicae

Configurations of points in sets of positive measure and in Baire sets of second category

François Destrempes, Ambar Sengupta (1989)

Fundamenta Mathematicae

Continuous measures and analytic sets

R. Kaufman (1989)

Colloquium Mathematicae

Convolution semigroup of measures on compact noncommutative semigroups

Štefan Schwarz (1964)

Czechoslovak Mathematical Journal

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...

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