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Characterizations of Kurzweil-Henstock-Pettis integrable functions

L. Di Piazza, K. Musiał (2006)

Studia Mathematica

We prove that several results of Talagrand proved for the Pettis integral also hold for the Kurzweil-Henstock-Pettis integral. In particular the Kurzweil-Henstock-Pettis integrability can be characterized by cores of the functions and by properties of suitable operators defined by integrands.

Checkerboards, Lipschitz functions and uniform rectifiability.

Peter W. Jones, Nets Hawk Katz, Ana Vargas (1997)

Revista Matemática Iberoamericana

In his recent lecture at the International Congress [S], Stephen Semmes stated the following conjecture for which we provide a proof.Theorem. Suppose Ω is a bounded open set in Rn with n > 2, and suppose that B(0,1) ⊂ Ω, Hn-1(∂Ω) = M < ∞ (depending on n and M) and a Lipschitz graph Γ (with constant L) such that Hn-1(Γ ∩ ∂Ω) ≥ ε.Here Hk denotes k-dimensional Hausdorff measure and B(0,1) the unit ball in Rn. By iterating our proof we obtain a slightly stronger result which allows us...

Choquet integrals in potential theory.

David R. Adams (1998)

Publicacions Matemàtiques

This is a survey of various applications of the notion of the Choquet integral to questions in Potential Theory, i.e. the integral of a function with respect to a non-additive set function on subsets of Euclidean n-space, capacity. The Choquet integral is, in a sense, a nonlinear extension of the standard Lebesgue integral with respect to the linear set function, measure. Applications include an integration principle for potentials, inequalities for maximal functions, stability for solutions to...

Choquet simplexes whose set of extreme points is K -analytic

Michel Talagrand (1985)

Annales de l'institut Fourier

We construct a Choquet simplex K whose set of extreme points T is 𝒦 -analytic, but is not a 𝒦 -Borel set. The set T has the surprising property of being a K σ δ set in its Stone-Cech compactification. It is hence an example of a K σ δ set that is not absolute.

Choquet-like integrals with respect to level-dependent capacities and ϕ -ordinal sums of aggregation function

Radko Mesiar, Peter Smrek (2015)

Kybernetika

In this study we merge the concepts of Choquet-like integrals and the Choquet integral with respect to level dependent capacities. For finite spaces and piece-wise constant level-dependent capacities our approach can be represented as a ϕ -ordinal sum of Choquet-like integrals acting on subdomains of the considered scale, and thus it can be regarded as extension method. The approach is illustrated by several examples.

Classes of fuzzy measures and distortion

Ľubica Valášková, Peter Struk (2005)

Kybernetika

Distortion of fuzzy measures is discussed. A special attention is paid to the preservation of submodularity and supermodularity, belief and plausibility. Full characterization of distortion functions preserving the mentioned properties of fuzzy measures is given.

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