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Marczewski-Burstin-like characterizations of σ-algebras, ideals, and measurable functions

Jack Brown, Hussain Elalaoui-Talibi (1999)

Colloquium Mathematicae

ℒ denotes the Lebesgue measurable subsets of ℝ and ℒ 0 denotes the sets of Lebesgue measure 0. In 1914 Burstin showed that a set M ⊆ ℝ belongs to ℒ if and only if every perfect P ∈ ℒ$ℒ0 h a s a p e r f e c t s u b s e t Q ∈ ℒ $ ℒ 0 which is a subset of or misses M (a similar statement omitting “is a subset of or” characterizes ℒ 0 ). In 1935, Marczewski used similar language to define the σ-algebra (s) which we now call the “Marczewski measurable sets” and the σ-ideal ( s 0 ) which we call the “Marczewski null sets”. M ∈ (s) if every perfect set P has...

Markov operators acting on Polish spaces

Tomasz Szarek (1997)

Annales Polonici Mathematici

We prove a new sufficient condition for the asymptotic stability of Markov operators acting on measures. This criterion is applied to iterated function systems.

Mass transport problem and derivation

Nacereddine Belili, Henri Heinich (1999)

Applicationes Mathematicae

A characterization of the transport property is given. New properties for strongly nonatomic probabilities are established. We study the relationship between the nondifferentiability of a real function f and the fact that the probability measure λ f * : = λ ◦ ( f * ) - 1 , where f*(x):=(x,f(x)) and λ is the Lebesgue measure, has the transport property.

Matchings and the variance of Lipschitz functions

Franck Barthe, Neil O'Connell (2009)

ESAIM: Probability and Statistics

We are interested in the rate function of the moderate deviation principle for the two-sample matching problem. This is related to the determination of 1-Lipschitz functions with maximal variance. We give an exact solution for random variables which have normal law, or are uniformly distributed on the Euclidean ball.

Matrix-Variate Statistical Distributions and Fractional Calculus

Mathai, A., Haubold, H. (2011)

Fractional Calculus and Applied Analysis

MSC 2010: 15A15, 15A52, 33C60, 33E12, 44A20, 62E15 Dedicated to Professor R. Gorenflo on the occasion of his 80th birthdayA connection between fractional calculus and statistical distribution theory has been established by the authors recently. Some extensions of the results to matrix-variate functions were also considered. In the present article, more results on matrix-variate statistical densities and their connections to fractional calculus will be established. When considering solutions of fractional...

Maximal distributional chaos of weighted shift operators on Köthe sequence spaces

Xinxing Wu (2014)

Czechoslovak Mathematical Journal

During the last ten some years, many research works were devoted to the chaotic behavior of the weighted shift operator on the Köthe sequence space. In this note, a sufficient condition ensuring that the weighted shift operator B w n : λ p ( A ) → λ p ( A ) defined on the Köthe sequence space λ p ( A ) exhibits distributional ϵ -chaos for any 0 < ϵ < diam λ p ( A ) and any n ∈ ℕ is obtained. Under this assumption, the principal measure of B w n is equal to 1. In particular, every Devaney chaotic shift operator exhibits distributional ϵ -chaos for any 0 < ϵ < diam λ p ( A ) .

Maximal scrambled sets for simple chaotic functions.

Víctor Jiménez López (1996)

Publicacions Matemàtiques

This paper is a continuation of [1], where a explicit description of the scrambled sets of weakly unimodal functions of type 2∞ was given. Its aim is to show that, for an appropriate non-trivial subset of the above family of functions, this description can be made in a much more effective and informative way.

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