Weak Comparative Probability on Infinite Sets

Abstract

Let $\mathscr{J}$ be a Boolean algebra of subsets of a state space $S$ and let $\succ$ be a binary comparative probability relation on $\mathscr{J}$ with $A \succ B$ interpreted as "$A$ is more probable than $B$." Axioms are given for $\succ$ on $\mathscr{J}$ which are sufficient for the existence of a finitely additive probability measure $P$ on $\mathscr{J}$ which has $P(A) > P(B)$ whenever $A \succ B$. The axioms consist of a necessary cancellation or additivity condition, a simple monotonicity axiom, an axiom for the preservation of $\succ$ under common deletions, and an Archimedean condition.