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Order convergence of vector measures on topological spaces

Surjit Singh Khurana (2008)

Mathematica Bohemica

Let X be a completely regular Hausdorff space, E a boundedly complete vector lattice, C b ( X ) the space of all, bounded, real-valued continuous functions on X , ℱ the algebra generated by the zero-sets of X , and μ C b ( X ) → E a positive linear map. First we give a new proof that μ extends to a unique, finitely additive measure μ ℱ → E + such that ν is inner regular by zero-sets and outer regular by cozero sets. Then some order-convergence theorems about nets of E + -valued finitely additive measures on ℱ are proved, which extend...

Order-theoretic properties of some sets of quasi-measures

Zbigniew Lipecki (2017)

Commentationes Mathematicae Universitatis Carolinae

Let 𝔐 and ℜ be algebras of subsets of a set Ω with 𝔐 ⊂ ℜ , and denote by E ( μ ) the set of all quasi-measure extensions of a given quasi-measure μ on 𝔐 to ℜ . We show that E ( μ ) is order bounded if and only if it is contained in a principal ideal in b a ( ℜ ) if and only if it is weakly compact and extr E ( μ ) is contained in a principal ideal in b a ( ℜ ) . We also establish some criteria for the coincidence of the ideals, in b a ( ℜ ) , generated by E ( μ ) and extr E ( μ ) .

p -symmetric bi-capacities

Pedro Miranda, Michel Grabisch (2004)

Kybernetika

Bi-capacities have been recently introduced as a natural generalization of capacities (or fuzzy measures) when the underlying scale is bipolar. They allow to build more flexible models in decision making, although their complexity is of order 3 n , instead of 2 n for fuzzy measures. In order to reduce the complexity, the paper proposes the notion of p -symmetric bi- capacities, in the same spirit as for p -symmetric fuzzy measures. The main idea is to partition the set of criteria (or states of nature,...

Packing spectra for Bernoulli measures supported on Bedford-McMullen carpets

Thomas Jordan, Michał Rams (2015)

Fundamenta Mathematicae

We consider the packing spectra for the local dimension of Bernoulli measures supported on Bedford-McMullen carpets. We show that typically the packing dimension of the regular set is smaller than the packing dimension of the attractor. We also consider a specific class of measures for which we are able to calculate the packing spectrum exactly, and we show that the packing spectrum is discontinuous as a function on the space of Bernoulli measures.

Parabolic Cantor sets

Mariusz Urbański (1996)

Fundamenta Mathematicae

The notion of a parabolic Cantor set is introduced allowing in the definition of hyperbolic Cantor sets some fixed points to have derivatives of modulus one. Such difference in the assumptions is reflected in geometric properties of these Cantor sets. It turns out that if the Hausdorff dimension of this set is denoted by h, then its h-dimensional Hausdorff measure vanishes but the h-dimensional packing measure is positive and finite. This latter measure can also be dynamically characterized as the...

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