## The Annals of Probability

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

### Amarts Indexed by Directed Sets

#### 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