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0-1 laws of a probability measure on a locally convex space. (English) Zbl 0605.60040
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.
Reviewer: D.Plachky

##### MSC:
 60F20 Zero-one laws 60B05 Probability measures on topological spaces
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##### References:
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