Preorderings compatible with probability measures

Chuaqui, Rolando; Malitz, Jerome

doi:10.1090/S0002-9947-1983-0709585-3

Preorderings compatible with probability measures
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by Rolando Chuaqui and Jerome Malitz PDF

Trans. Amer. Math. Soc. 279 (1983), 811-824 Request permission

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 $B’$ admits a probability measure $\mu ’$ such that for $p,q \in B’,p \succcurlyeq q\;iff\mu ’p \geqslant \mu ’q$. (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 $q’ \subseteq q$ with $q’ \preccurlyeq q,q’ \npreceq 0$, and $q \npreceq q’$, 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 $q’ \subseteq q$ implies $q \preccurlyeq q’\;or\;q’ \preccurlyeq 0$, then we have that for every $p,q \in B,p \succcurlyeq q$ implies $\mu p \geqslant \mu q$.

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