Continuity of quasi-invariant measures and zero-one laws on groups
type: Item
instance of: scholarly article
title: Continuity of quasi-invariant measures and zero-one laws on groups
zbMATH Open document ID: 0812.60036
author: Hitoshi Mizumachi
author: Hiroshi Sato
published in: Journal of Functional Analysis
publication date: 1994-07-04T00:00:00Z
review text: Let \((X,{\mathcal B},\mu)\) be a probability space, and let \(G\) be a measurable action group on \(X\). The first aim of this paper is to give several equivalent conditions for \(\mu \ll m_ G*\mu\), where \(m_ G\) stands for the left Haar measure on \(G\). Another interesting result is as follows: Let \({\mathbf X} = (X_ n)_{n \geq 1}\) be a real random sequence, let \(\mu\) be the distribution of \({\mathbf X}\), and let \(\mu_ n\) stand for the distribution of \((X_ 1, \dots, X_ n)\). If the tail \(\sigma\)-algebra of \({\mathbf X}\) is trivial, and \(\mu_ n \ll \lambda_ n\), where \(\lambda_ n\) denotes the Lebesgue measure on \(R^ n\), then \(\mu (H+x) = 0\) or 1, \(x \in R^ \infty\), for each \(\mu\)-measurable subgroup \(H\) of \(R^ \infty\).
Mathematics Subject Classification ID: 60F20
Mathematics Subject Classification ID: 60B15
zbMATH DE Number: 599755
zbMATH Keywords: zero-one laws on groups
zbMATH Keywords: measurable action group
zbMATH Keywords: Haar measure
reviewed by: Aurel Spătaru
MaRDI profile type: Publication
full work available at URL: jfan.1994.1029
OpenAlex ID: W1964009013
DOI: 10.1006/JFAN.1994.1029
description: scientific article; zbMATH DE number 599755