Preorderings compatible with probability measures

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 $\mu p \geqslant \mu q$.