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April, 1978 Amarts Indexed by Directed Sets
Kenneth A. Astbury
Ann. Probab. 6(2): 267-278 (April, 1978). DOI: 10.1214/aop/1176995572

Abstract

We prove that an amart indexed by a directed set decomposes into a martingale and an amart which converges to zero in $L_1$ norm. The main theorem asserts that the underlying family of $\sigma$-algebras satisfies the Vitali condition if and only if every $L_1$ bounded amart essentially converges.

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Kenneth A. Astbury. "Amarts Indexed by Directed Sets." Ann. Probab. 6 (2) 267 - 278, April, 1978. https://doi.org/10.1214/aop/1176995572

Information

Published: April, 1978
First available in Project Euclid: 19 April 2007

zbMATH: 0378.60017
MathSciNet: MR464394
Digital Object Identifier: 10.1214/aop/1176995572

Subjects:
Primary: 60F15
Secondary: 46G10 , 60G40 , 60G45 , 60G99

Keywords: Amart , directed set , essential convergence , martingale , potential , Vitali condition

Rights: Copyright © 1978 Institute of Mathematical Statistics

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Vol.6 • No. 2 • April, 1978
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