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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Preorderings compatible with probability measures
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by Rolando Chuaqui and Jerome Malitz
Trans. Amer. Math. Soc. 279 (1983), 811-824
DOI: https://doi.org/10.1090/S0002-9947-1983-0709585-3

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 $\mu p \geqslant \mu q$.
References
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Bibliographic Information
  • © Copyright 1983 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 279 (1983), 811-824
  • MSC: Primary 60A10; Secondary 03H05
  • DOI: https://doi.org/10.1090/S0002-9947-1983-0709585-3
  • MathSciNet review: 709585