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Density in the space of topological measures

S. V. Butler (2002)

Fundamenta Mathematicae

Topological measures (formerly "quasi-measures") are set functions that generalize measures and correspond to certain non-linear functionals on the space of continuous functions. The goal of this paper is to consider relationships between various families of topological measures on a given space. In particular, we prove density theorems involving classes of simple, representable, extreme topological measures and measures, hence giving a way of approximating various topological measures by members...

Density of a family of linear varietes

Grazia Raguso, Luigia Rella (2006)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

The measurability of the family, made up of the family of plane pairs and the family of lines in 3 -dimensional space A 3 , is stated and its density is given.

Derivability, variation and range of a vector measure

L. Rodríguez-Piazza (1995)

Studia Mathematica

We prove that the range of a vector measure determines the σ-finiteness of its variation and the derivability of the measure. Let F and G be two countably additive measures with values in a Banach space such that the closed convex hull of the range of F is a translate of the closed convex hull of the range of G; then F has a σ-finite variation if and only if G does, and F has a Bochner derivative with respect to its variation if and only if G does. This complements a result of [Ro] where we proved...

Derivative of the Donsker delta functionals

Herry Pribawanto Suryawan (2019)

Mathematica Bohemica

We prove that derivatives of any finite order of Donsker's delta functionals are well-defined elements in the space of Hida distributions. We also show the convergence to the derivative of Donsker's delta functionals of two different approximations. Finally, we present an existence result of finite product and infinite series of the derivative of the Donsker delta functionals.

Descriptive properties of mappings between nonseparable Luzin spaces

Petr Holický, Václav Komínek (2007)

Czechoslovak Mathematical Journal

We relate some subsets G of the product X × Y of nonseparable Luzin (e.g., completely metrizable) spaces to subsets H of ℕ ℕ × Y in a way which allows to deduce descriptive properties of G from corresponding theorems on H . As consequences we prove a nonseparable version of Kondô’s uniformization theorem and results on sets of points y in Y with particular properties of fibres f - 1 ( y ) of a mapping f X → Y . Using these, we get descriptions of bimeasurable mappings between nonseparable Luzin spaces in terms of fibres.

Descriptive set-theoretical properties of an abstract density operator

Szymon Gła̧b (2009)

Open Mathematics

Let 𝒦 (ℝ) stand for the hyperspace of all nonempty compact sets on the real line and let d ±(x;E) denote the (right- or left-hand) Lebesgue density of a measurable set E ⊂ ℝ at a point x∈ ℝ. In [3] it was proved that { K ∈ 𝒦 ( ℝ ) : ∀ x ∈ K ( d + ( x , K ) = 1 o r d - ( x , K ) = 1 ) } is ⊓11-complete. In this paper we define an abstract density operator ⅅ± and we generalize the above result. Some applications are included.

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