The Annals of Probability

Amarts Indexed by Directed Sets

Kenneth A. Astbury

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.

Article information

Source
Ann. Probab., Volume 6, Number 2 (1978), 267-278.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176995572

Digital Object Identifier
doi:10.1214/aop/1176995572

Mathematical Reviews number (MathSciNet)
MR464394

Zentralblatt MATH identifier
0378.60017

JSTOR