0-1 laws of a probability measure on a locally convex space
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 semi-norm N(x) on E (admitting the value \(\infty)\), \(\mu (x;N(x)<\infty)=0\) or 1. (15) For every compact convex balanced subset K, \(\mu\) (\(\cup^{\infty}_{n=1}nK)=0\) or 1. Main results: Theorem 1: The 0-1 laws (2)-(11) are equivalent. Theorem 2: Suppose that \(\mu\) is a Radon probability measure. Then the 0-1 laws (2)- (11), (13) and (14) are all equivalent. Theorem 3: If \(\mu\) is a convex Radon measure, then the 0-1 laws (2)-(11), (13), (14) and (15) are all equivalent.
scientific article; zbMATH DE number 3978049