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Displaying 461 – 480 of 2107

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Convergence theorems for the PU-integral

Giuseppa Riccobono (2000)

Mathematica Bohemica

We give a definition of uniform PU-integrability for a sequence of μ -measurable real functions defined on an abstract metric space and prove that it is not equivalent to the uniform μ -integrability.

Coordinatewise decomposition of group-valued Borel functions

Benjamin D. Miller (2007)

Fundamenta Mathematicae

Answering a question of Kłopotowski, Nadkarni, Sarbadhikari, and Srivastava, we characterize the Borel sets S ⊆ X × Y with the property that every Borel function f: S → ℂ is of the form f(x,y) = u(x) + v(y), where u: X → ℂ and v: Y → ℂ are Borel.

Correlation dimension for self-similar Cantor sets with overlaps

Károly Simon, Boris Solomyak (1998)

Fundamenta Mathematicae

We prove a classification theorem of the “Glimm-Effros” type for Borel order relations: a Borel partial order on the reals either is Borel linearizable or includes a copy of a certain Borel partial order 0 which is not Borel linearizable.

Corrigendum to (n,2)-sets have full Hausdorff dimension.

Themis Mitsis (2005)

Revista Matemática Iberoamericana

In the paper (n,2)-sets have full Hausdorff dimension, appeared in Rev. Mat. Iberoamericana 20 (2004), 381-393, the author claimed that an (n,2)-set must have full Hausdorff dimension. However, as pointed out by Terence Tao and John Bueti, the proof contains an error.

Countable contraction mappings in metric spaces: invariant sets and measure

María Barrozo, Ursula Molter (2014)

Open Mathematics

We consider a complete metric space (X, d) and a countable number of contraction mappings on X, F = {F i: i ∈ ℕ}. We show the existence of a smallest invariant set (with respect to inclusion) for F. If the maps F i are of the form F i(x) = r i x + b i on X = ℝd, we prove a converse of the classic result on contraction mappings, more precisely, there exists a unique bounded invariant set if and only if r = supi r i is strictly smaller than 1. Further, if ρ = {ρ k}k∈ℕ is a probability sequence, we...

Countable tightness in the spaces of regular probability measures

Grzegorz Plebanek, Damian Sobota (2015)

Fundamenta Mathematicae

We prove that if K is a compact space and the space P(K × K) of regular probability measures on K × K has countable tightness in its weak* topology, then L₁(μ) is separable for every μ ∈ P(K). It has been known that such a result is a consequence of Martin's axiom MA(ω₁). Our theorem has several consequences; in particular, it generalizes a theorem due to Bourgain and Todorčević on measures on Rosenthal compacta.

Covering the real line with translates of a zero-dimensional compact set

András Máthé (2011)

Fundamenta Mathematicae

We construct a compact set C of Hausdorff dimension zero such that cof(𝒩) many translates of C cover the real line. Hence it is consistent with ZFC that less than continuum many translates of a zero-dimensional compact set can cover the real line. This answers a question of Dan Mauldin.

Curvature bounds for neighborhoods of self-similar sets

Steffen Winter (2011)

Commentationes Mathematicae Universitatis Carolinae

In some recent work, fractal curvatures C k f ( F ) and fractal curvature measures C k f ( F , · ) , k = 0 , ... , d , have been determined for all self-similar sets F in d , for which the parallel neighborhoods satisfy a certain regularity condition and a certain rather technical curvature bound. The regularity condition is conjectured to be always satisfied, while the curvature bound has recently been shown to fail in some concrete examples. As a step towards a better understanding of its meaning, we discuss several equivalent formulations...

Currently displaying 461 – 480 of 2107