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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)


Preorderings compatible with probability measures

Authors: Rolando Chuaqui and Jerome Malitz
Journal: Trans. Amer. Math. Soc. 279 (1983), 811-824
MSC: Primary 60A10; Secondary 03H05
MathSciNet review: 709585
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Abstract: The main theorem proved in this paper is:

Let $ B$ be a $ \sigma $-complete Boolean algebra and $ \succcurlyeq a$ binary relation with field $ B$ such that:

(i) Every finite subalgebra $ B^{\prime}$ admits a probability measure $ \mu^{\prime}$ such that for $ p,q \in B^{\prime},p \succcurlyeq q\;iff\mu 'p \geqslant \mu 'q$.

(ii) If for every $ i,{p_i},q \in B$ and $ {p_i} \subseteq {p_{i + 1}} \preccurlyeq q$, then $ { \cup_{i < \infty }}{p_i} \preccurlyeq q$.

Under these conditions there is a $ \sigma $-additive probability measure $ \mu $ on $ B$ such that:

(a) If there is $ a\;p \in B$, such that for every $ q \subseteq p$ there is a $ q^{\prime} \subseteq q$ with $ q^{\prime} \preccurlyeq q,q^{\prime} \npreceq 0$, and $ q \npreceq q^{\prime}$, then we have that for every $ p,q \in B,\mu \,p \geqslant \mu \,q\,iff\,p \succcurlyeq q$.

(b) If for every $ p \in B$, there is $ a\;q \subseteq p$ such that $ q^{\prime} \subseteq q$ implies $ q \preccurlyeq q^{\prime}\;or\;q^{\prime} \preccurlyeq 0$, then we have that for every $ p,q \in B,p \succcurlyeq q$ implies $ \mu p \geqslant \mu q$.

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Additional Information

PII: S 0002-9947(1983)0709585-3
Article copyright: © Copyright 1983 American Mathematical Society

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