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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...

Cardinality of some convex sets and of their sets of extreme points

Zbigniew Lipecki (2011)

Colloquium Mathematicae

We show that the cardinality of a compact convex set W in a topological linear space X satisfies the condition that ℵ ₀ = . We also establish some relations between the cardinality of W and that of extrW provided X is locally convex. Moreover, we deal with the cardinality of the convex set E(μ) of all quasi-measure extensions of a quasi-measure μ, defined on an algebra of sets, to a larger algebra of sets, and relate it to the cardinality of extrE(μ).

Category bases.

Detlefsen, M., Szymański, Andrzej (1993)

International Journal of Mathematics and Mathematical Sciences

Category measures on Baire spaces.

José María Ayerbe Toledano (1990)

Publicacions Matemàtiques

The purpose of this paper is to give a necessary and sufficient condition to define a category measure on a Baire topological space. In the last section we give some examples of spaces in these conditions.

Central limit theorems for non-invertible measure preserving maps

Michael C. Mackey, Marta Tyran-Kamińska (2008)

Colloquium Mathematicae

Using the Perron-Frobenius operator we establish a new functional central limit theorem for non-invertible measure preserving maps that are not necessarily ergodic. We apply the result to asymptotically periodic transformations and give a specific example using the tent map.

Chaotic behavior of infinitely divisible processes

S. Cambanis, K. Podgórski, A. Weron (1995)

Studia Mathematica

The hierarchy of chaotic properties of symmetric infinitely divisible stationary processes is studied in the language of their stochastic representation. The structure of the Musielak-Orlicz space in this representation is exploited here.

Characteristic points, rectifiability and perimeter measure on stratified groups

Valentino Magnani (2006)

Journal of the European Mathematical Society

We establish an explicit connection between the perimeter measure of an open set E with C 1 boundary and the spherical Hausdorff measure S Q − 1 restricted to ∂ E , when the ambient space is a stratified group endowed with a left invariant sub-Riemannian metric and Q denotes the Hausdorff dimension of the group. Our formula implies that the perimeter measure of E is less than or equal to S Q − 1 ( ∂ E ) up to a dimensional factor. The validity of this estimate positively answers a conjecture raised by Danielli, Garofalo...

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