Uniform separation of points and measures and representation by sums of algebras
type: Item
instance of: scholarly article
title: Uniform separation of points and measures and representation by sums of algebras
zbMATH Open document ID: 0628.46017
DOI: 10.1007/BF02765032
published in: Israel Journal of Mathematics
publication date: 1986-01-01T00:00:00Z
review text: Let X and \(Y_ i\), \(1\leq i\leq k\), be compact metric spaces, and let \(\rho_ i: X\to Y_ i\) be continuous functions. The family \(F=\{\rho_ i\}^ k_{i=1}\) is said to be a measure separating family, if there exists some \(\lambda >0\) such that for every measure \(\mu\) in \(C(X)^*\), \(\| \mu \circ \phi_ i^{-1}\| \geq \lambda \| \mu \|\) holds for some \(1\leq i\leq k\). F is a uniformly (point) separating family if the above holds for the purely atomic measures in \(C(X)^*\). It is known that for \(k\leq 2\) the two concepts are equivalent. In this note we present examples which show that for \(k\geq 3\) measure separation is a stronger property than uniform separation of points, and characterize those uniformly separating families which separate measures. These properties and problems are closely related to the following ones: let \(A_ 1,A_ 2,...,A_ k\) be closed subalgebras of C(X); when is \(A_ 1+A_ 2+...+A_ k\) equal to or dense in C(X)?
Mathematics Subject Classification ID: 46E27
Mathematics Subject Classification ID: 46J10
zbMATH DE Number: 4021891
zbMATH Keywords: measure separating family
zbMATH Keywords: purely atomic measures
zbMATH Keywords: uniform separation of points
author: Yaki Sternfeld
MaRDI profile type: Publication
cites work: Q3887567
cites work: Approximation by a sum of two algebras. The lightning bolt principle
cites work: Dimension of metric spaces and Hilbert’s problem 13
cites work: Uniformly separating families of functions
cites work: Superpositions of continuous functions
full work available at URL: https://doi.org/10.1007/bf02765032
OpenAlex ID: W1994566194
description: scientific article; zbMATH DE number 4021891