0-1 laws of a probability measure on a locally convex space
type: Item
instance of: scholarly article
title: 0-1 laws of a probability measure on a locally convex space
zbMATH Open document ID: 0605.60040
author: Yasuji Takahashi
author: Yoshiaki Okazaki
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1986-01-01T00:00:00Z
review text: The authors introduce the following notions of 0-1 laws for a measure \(\mu\) defined on the \(\sigma\)-algebra generated by the topological dual E' of a locally convex Hausdorff space E resp. on the \(\sigma\)-algebra generated by the open subsets of E. (0) For every x'\(\in E'\), \(\mu (x;x'(x)=0)=0\) or 1. (1) p5 For every \(\mu\)-measurable linear subspace \(F\subset E\), \(\mu (F)=0\) or 1. (2) For every sequence \(x_ n'\in E'\) and every convex Lusin subspace F of \(R^{\infty}\), \(\mu (x;(x_ n'(x))\in F)=0\) or 1. (3) For every sequence \(x_ n'\in E'\), \(\mu (x;(x_ n'(x))\in l_{\infty})=0\) or 1. (4) For every sequence \(x_ n'\in E'\), \(\mu (x;(x_ n'(x))\in c_ 1)=0\) or 1. (5) For every sequence \(x_ n'\in E'\), \(\mu (x;(x_ n'(x))\in c_ 0)=0\) or 1. (6) There exist no sequence \(x_ n'\in E'\) such that \(\mu (x;(x_ n'(x))\in c_ 0)>0\) and that \(\mu (x;(x_ n'(x)\not\in l_{\infty})>0.\) (7) For every sequence \(x_ n'\in E'\), if there exists \((a_ n)\in c_ 0\) such that \(\mu (x;(a_ n^{-1}x_ n'(x))\in l_{\infty})>0\), then \(\mu (x;(x_ n'(x))\in c_ 0)=1.\) (8) For every sequence \(x_ n'\in E'\), if there exists \((a_ n)\in c_ 0\) such that \(\mu (x;(a_ n^{-1}x_ n'(x))\in l_{\infty})>0\), then \(\mu (x;(x_ n'(x)\in l_{\infty})=1.\) (9) For every sequence \(x_ n'\in E'\), if there exists \((a_ n)\in c_ 0\) such that \(\mu (x;(a_ n^{-1}x_ n'(x))\in c_ 1)>0\), then \(\mu (x;(x_ n'(x))\in c_ 1)=1.\) (10) For every sequence \(x_ n'\in E'\), if there exists \((a_ n)\in c_ 0\) such that \(\mu (x;(a_ n^{-1}x_ n'(x))\in c_ 0)>0\), then \(\mu (x;(x_ n'(x))\in c_ 0)=1.\) (11) For every sequence \(x_ n'\in E'\), if there exists \((a_ n)\in c_ 0\) such that \(\mu (x;(a_ n^{-1}x_ n'(x))\in c_ 0)>0\), then \(\mu (x;(x_ n'(x))\in l_{\infty})=1.\) (12) For every sequence \(x_ n'\in E'\), \(\mu (x;(x_ n'(x))\in l_ p)=0\) or 1 \((1\leq p<\infty).\) (13) For every closed convex balanced subset B, \(\mu\) (\(\cup^{\infty}_{n=1}nB)=0\) or 1. (14) For every lower semi-continuous s…
Mathematics Subject Classification ID: 60F20
Mathematics Subject Classification ID: 60B05
zbMATH DE Number: 3978049
zbMATH Keywords: Radon measure
zbMATH Keywords: convex Radon measure
zbMATH Keywords: locally convex Hausdorff space
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195178374
OpenAlex ID: W1977884084
Wikidata QID: Q114845158
cites work: Convex measures on locally convex spaces
cites work: Zero-One Laws for Stable Measures
cites work: Integrability of seminorms, the 0-1 law and the affine kernel for product measures
cites work: Q4194205
cites work: Q3948373
cites work: Q4113166
cites work: Hilbertian support of probability measures on locally convex spaces and their applications
cites work: Zero-One Laws for Non-Gaussian Measures
DOI: 10.2977/PRIMS/1195178374
description: scientific article; zbMATH DE number 3978049