An aspect of differentiable measures on \({\mathbb{R}}^{\infty}\)
type: Item
instance of: scholarly article
title: An aspect of differentiable measures on \({\mathbb{R}}^{\infty}\)
zbMATH Open document ID: 0651.60005
author: Hiroaki Shimomura
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1987-01-01T00:00:00Z
review text: Let \(R^{\infty}\) be a countable direct product of \(R^ 1\), \({\mathcal B}(R^{\infty})\) be the usual Borel \(\sigma\)-algebra on \(R^{\infty}\) and \(\mu\) be a probability measure on \({\mathcal B}(R^{\infty})\). \(\mu\) is said to be differentiable in the a-direction, or a is said to be a differentiable shift, if \[ \lim_{h\to 0}h^{-1}\{\mu (E-ta)-\mu (E)\}\equiv \partial_ a\mu (E) \] exists for all \(E\in {\mathcal B}(R^{\infty})\). In this case \(\partial_ a\mu\) is absolutely continuous with respect to \(\mu\). So the set \(D_{\mu}\) of all differentiable shifts for \(\mu\) equipped with the norm \[ \| a\|_{\mu}=\int | d\partial_ a\mu /d\mu (x)| d\mu (x) \] is a Banach space of cotype 2. We investigate \(R_ 0^{\infty}\)- differentiable measures \(\mu\), especially direct-product measures \(\prod_{n}\mu_ n\) of \(\mu_ n\), being 1-dimensional probability measures on \({\mathcal B}(R^ 1)\), where \[ R_ 0^{\infty}=\{a=(a_ n)\in R^{\infty}| a_ n=0\text{ except for a finite number of n's}\}. \] The following main results are obtained. (1) Let \(\mu =\prod_{n}\mu_ n\) be \(R_ 0^{\infty}\)-differentiable on \(R^ 1\), \(\mu_ n\) \((n=1,...)\) have the density \(f_ n(x)\) with respect to the Lebesgue measure dx on \(R^ 1\) and \(f_ n'(x)\) belong to \(L^ 1_{dx}(R^ 1)\). Put \(\phi_ n(x)=f_ n'/f_ n(x)\). Then \[ D_{\mu}=\{a=(a_ n)\in R^{\infty}| \sum_{n}M_ n(a_ n)<\infty \}, \] where \(M_ n\) \((n=1,...)\) is an Orlicz function on \(R^ 1\) defined by \[ M_ n(\alpha)=\int^{\infty}_{- \infty}dx\int^{1}_{0}\{1-\exp (-2^{-1}\alpha^ 2u^ 2\phi^ 2_ n(x))\}u^{-2}du. \] Thus \(D_{\mu}\) is a modular sequence space. (2) In particular, if \(\mu\) is a stationary product, that is \(\mu_ n\) is the same measure, say \(\mu_ 0\) for all n, then \(D_{\mu}\) is an Orlicz sequence space defined by the function \(M_ 0(\alpha)\). So we can give necessary and sufficient conditions for the problem if there exists some \(\mu\) such that \(D_{\mu}=\ell_ M\), or we can give a characterization…
reviewed by: Hiroaki Shimomura
Mathematics Subject Classification ID: 60B11
Mathematics Subject Classification ID: 28C20
zbMATH DE Number: 4062228
zbMATH Keywords: measures on infinite dimensional spaces
zbMATH Keywords: differentiable measure
zbMATH Keywords: differentiable shift
zbMATH Keywords: Orlicz function
zbMATH Keywords: Orlicz sequence
zbMATH Keywords: translational- quasi-invariance
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195176033
OpenAlex ID: W905764886
cites work: Q4136031
cites work: An aspect of quasi-invariant measures on \(R^\infty\)
cites work: Some results on quasi-invariant measures on infinite-dimensional spaces
cites work: Ergodic decomposition of quasi-invariant measures
cites work: Differentiable shifts for measures on infinite dimensional spaces
DOI: 10.2977/PRIMS/1195176033
description: scientific article; zbMATH DE number 4062228