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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$.

References [Enhancements On Off] (What's this?)

  • [KLST] D. H. Krantz, R. D. Luce, P. Suppes and A. Tversky, Foundations of measurement, Vol. I, Academic Press, New York, 1971. MR 0459067 (56:17265)
  • [V] C. Villegas, On qualitative probability $ \sigma $-algebras, Ann. Math. Statist. 35 (1964), 1787-1796. MR 0167588 (29:4860)
  • [SC] M. G. Schwarze and R. Chuaqui, $ \sigma $-additive measurement structures, Mathematical Logic in Latin America (Arruda, Chuaqui and da Costa, eds.), North-Holland, Amsterdam, 1980, pp. 351-364. MR 573956 (81g:90009)
  • [KPS] C. H. Kraft, J. W. Pratt and A. Seidenberg, Intuitive probability on finite sets, Ann. Math. Statist. 30 (1959), 408-419. MR 0102850 (21:1636)

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Article copyright: © Copyright 1983 American Mathematical Society

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