## The Annals of Statistics

### Agreeing Probability Measures for Comparative Probability Structures

Peter Wakker

#### Abstract

It is proved that fine and tight comparative probability structures (where the set of events is assumed to be an algebra, not necessarily a $\sigma$-algebra) have agreeing probability measures. Although this was often claimed in the literature, all proofs the author encountered are not valid for the general case, but only for $\sigma$-algebras. Here the proof of Niiniluoto (1972) is supplemented. Furthermore an example is presented that reveals many misunderstandings in the literature. At the end a necessary and sufficient condition is given for comparative probability structures to have an almost agreeing probability measure.

#### Article information

Source
Ann. Statist., Volume 9, Number 3 (1981), 658-662.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176345469

Digital Object Identifier
doi:10.1214/aos/1176345469

Mathematical Reviews number (MathSciNet)
MR615441

Zentralblatt MATH identifier
0474.60004

JSTOR
links.jstor.org

Subjects
Primary: 60A05: Axioms; other general questions
Secondary: 92A25 06A05: Total order

#### Citation

Wakker, Peter. Agreeing Probability Measures for Comparative Probability Structures. Ann. Statist. 9 (1981), no. 3, 658--662. doi:10.1214/aos/1176345469. https://projecteuclid.org/euclid.aos/1176345469