On the approximation of measurable linear functionals
type: Item
instance of: scholarly article
title: On the approximation of measurable linear functionals
zbMATH Open document ID: 0582.60005
DOI: 10.1016/0167-7152(85)90019-7
author: B. George
published in: Statistics \& Probability Letters
publication date: 1985-01-01T00:00:00Z
review text: Let \(\mu\) be a probability Radon measure on a locally convex space E. \(\mu\) is said to have the Riesz property if for every \(\mu\)-measurable linear functional f on E there exists a sequence of elements of E' converging to f in \(\mu\). It is proved that every Gaussian measure has the Riesz property. This extends the result of \textit{M. Kanter} [J. Funct. Anal. 9, 441-459 (1972; Zbl 0232.60023)] obtained in the symmetric case. It is also proved that a shift of the law on \(R^{\infty}\) of an infinite Bernoulli sequence has the Riesz property iff it is not square summable or belongs to \(R_ 0^{\infty}.\) In a forthcoming paper of the author [to appear in Bull. Pol. Acad. Sci., Math.] the following generalization of the results mentioned above is proved: A shift of a symmetric Radon probability measure with the Riesz property inherits this property iff it is not contained in \(J_{\mu}\setminus SJ_{\mu}\), \(where\) SJ\({}_{\mu}=\cap \{L:\mu (L)=1\) and L is a linear \(set\};\) J\({}_{\mu}=\cap \{L:\mu (L)=1\), L is a linear set and \(L=\cup^{\infty}_{n=1}K_ n\), \(K_ n\) convex, compact\(\}\).
Mathematics Subject Classification ID: 60B05
zbMATH DE Number: 3932076
zbMATH Keywords: Riesz property
zbMATH Keywords: Gaussian measure
zbMATH Keywords: Bernoulli sequence
MaRDI profile type: Publication
full work available at URL: 0167-7152(85)90019-7
OpenAlex ID: W2056639911
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description: scientific article; zbMATH DE number 3932076