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Information Center for Mathematical Science

논문검색

Information Center for Mathematical Science

논문검색

Manuscripta Mathematica.
( Vol. 69 NO.1 / (1990))
A Simple Characterization of Almost Uniform Convergence by Stochastic Convergence
D. Plachky,
Pages. 27-30
Abstract Motivated by Egorov's theorem and the characterization of the equivalence of $p$-stochastic convergence and $P$-almost convergence by the property of the probability distribution $P$ to be purely atomic and concentrated on a countable number of pairwise disjoint $P$-atoms (cf. [1], p.68), it is proved that $P$-stochastic resp. $P$-almost convergence is equivalent to $P$-almost uniform convergence (cf. [2], p. 89/90) if and only if $P$ is purely atomic and concentrated on a finite number of pairwise disjoint $P$-atoms. Furthermore, this property of $P$ is equivalent to the condition that any $P$-stochastic convergent sequence admits a $P$-almost uniform convergent subsequence. Finally a proof is given that $P$ is purely atomic and concentrated on a finite number of pariwise disjoint $P$-atoms if and only if there does not exist a purely finitely additive ${0,1 }$-valued probability charge, which vanishes for all $P$-zero sets.
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