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Range of density measures

Martin Sleziak, Miloš Ziman (2009)

Acta Mathematica Universitatis Ostraviensis

We investigate some properties of density measures – finitely additive measures on the set of natural numbers extending asymptotic density. We introduce a class of density measures, which is defined using cluster points of the sequence A ( n ) n as well as cluster points of some other similar sequences. We obtain range of possible values of density measures for any subset of . Our description of this range simplifies the description of Bhashkara Rao and Bhashkara Rao [Bhaskara Rao, K. P. S., Bhaskara Rao,...

Rank and spectral multiplicity

Sébastien Ferenczi, Jan Kwiatkowski (1992)

Studia Mathematica

For a dynamical system (X,T,μ), we investigate the connections between a metric invariant, the rank r(T), and a spectral invariant, the maximal multiplicity m(T). We build examples of systems for which the pair (m(T),r(T)) takes values (m,m) for any integer m ≥ 1 or (p-1, p) for any prime number p ≥ 3.

Relations between Shy Sets and Sets of ν p -Measure Zero in Solovay’s Model

G. Pantsulaia (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

An example of a non-zero non-atomic translation-invariant Borel measure ν p on the Banach space p ( 1 p ) is constructed in Solovay’s model. It is established that, for 1 ≤ p < ∞, the condition " ν p -almost every element of p has a property P" implies that “almost every” element of p (in the sense of [4]) has the property P. It is also shown that the converse is not valid.

Remarks on the tightness of cocycles

Jon Aaronson, Benjamin Weiss (2000)

Colloquium Mathematicae

We prove a generalised tightness theorem for cocycles over an ergodic probability preserving transformation with values in Polish topological groups. We also show that subsequence tightness of cocycles over a mixing probability preserving transformation implies tightness. An example shows that this latter result may fail for cocycles over a mildly mixing probability preserving transformation.

Représentation intégrale de certaines mesures quasi-invariantes sur 𝒞 ( 𝐑 ) ; mesures extrémales et propriété de Markov

Gilles Royer, Marc Yor (1976)

Annales de l'institut Fourier

On établit pour le cône C des mesures μ positives bornées sur 𝒞 ( R ) , quasi-invariantes sous les translations de 𝒟 ( R ) et vérifiant : μ ( f + d w ) = μ ( d w ) exp R d t [ ( w ( t ) + 1 2 f ( t ) ) f ' ' ( t ) - P ( w ( t ) + f ( t ) + P ( w ( t ) ) ] (avec P polynôme borné inférieurement) les résultats suivants :– Toute mesure de C est intégrale de mesures appartenant aux génératrices extrémales de  C .– Les génératrices extrémales de C sont composées de mesures markoviennes.

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