The Annals of Probability

Amarts Indexed by Directed Sets

Kenneth A. Astbury

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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

Permanent link to this document
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
links.jstor.org

Subjects
Primary: 60F15: Strong theorems
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 60G45 60G99: None of the above, but in this section 46G10: Vector-valued measures and integration [See also 28Bxx, 46B22]

Keywords
Amart martingale potential directed set essential convergence Vitali condition

Citation

Astbury, Kenneth A. Amarts Indexed by Directed Sets. Ann. Probab. 6 (1978), no. 2, 267--278. doi:10.1214/aop/1176995572. https://projecteuclid.org/euclid.aop/1176995572


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