Differentiable shifts for measures on infinite dimensional spaces
type: Item
instance of: scholarly article
title: Differentiable shifts for measures on infinite dimensional spaces
zbMATH Open document ID: 0631.46042
author: Yasuo Yamasaki
author: Akihito Hora
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1987-01-01T00:00:00Z
review text: The purpose of this paper is to develop differentiation theory of measures on infinite dimensional vector space. Here, the following definition is adopted: Let X be a real vector space, the measure \(\mu\) is said to be differentiable for the shift a in X if: \(1/t(\mu_{ta}-\mu)\) (where \(\mu_{ta}\Doteq \mu (\cdot +ta))\) converges as \(t\to 0\) in \({\mathbb{R}}\), with respect to the so-called \(\tau_ s\)-topology, that is the one defined by the family of semi-norms \(\{| \mu (A)| /A\) measurable\(\}\). If the limit is denoted by \(\partial_ a\mu\), properties of the differentiable operator \(\partial\) are studied in a first chapter, for instance in relation with conditional and product measures. In a second chapter, the authors apply these theories to product measures on \({\mathbb{R}}^{\infty}\), equipped with the usual weak topology: characterization of \(R_ 0^{\infty}\)-differentiability is given and differentiability with respect to \(\ell^ 2\) is discussed in detail.
reviewed by: P.Raboin
Mathematics Subject Classification ID: 46G12
Mathematics Subject Classification ID: 46G05
Mathematics Subject Classification ID: 28C20
Mathematics Subject Classification ID: 28A35
zbMATH DE Number: 4026127
zbMATH Keywords: differentiation theory of measures on infinite dimensional vector space
zbMATH Keywords: product measures
zbMATH Keywords: weak topology
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195176541
OpenAlex ID: W2053577060
cites work: Q4185089
DOI: 10.2977/PRIMS/1195176541
description: scientific article; zbMATH DE number 4026127