The main theorem proved in this paper is: Let B B be a σ \sigma -complete Boolean algebra and ≽ a \succcurlyeq a binary relation with field B B such that: (i) Every finite subalgebra B ′ B’ admits a probability measure μ ′ \mu ’ such that for p , q ∈ B ′ , p ≽ q i f f μ ′ p ⩾ μ ′ q p,q \in B’,p \succcurlyeq q\;iff\mu ’p \geqslant \mu ’q . (ii) If for every i , p i , q ∈ B i,{p_i},q \in B and p i ⊆ p i + 1 ≼ q {p_i} \subseteq {p_{i + 1}} \preccurlyeq q , then ∪ i > ∞ p i ≼ q { \cup _{i > \infty }}{p_i} \preccurlyeq q . Under these conditions there is a σ \sigma -additive probability measure μ \mu on B B such that: (a) If there is a p ∈ B a\;p \in B , such that for every q ⊆ p q \subseteq p there is a q ′ ⊆ q q’ \subseteq q with q ′ ≼ q , q ′ ⋠ 0 q’ \preccurlyeq q,q’ \npreceq 0 , and q ⋠ q ′ q \npreceq q’ , then we have that for every p , q ∈ B , μ p ⩾ μ q i f f p ≽ q p,q \in B,\mu \,p \geqslant \mu \,q\,iff\,p \succcurlyeq q . (b) If for every p ∈ B p \in B , there is a q ⊆ p a\;q \subseteq p such that q ′ ⊆ q q’ \subseteq q implies q ≼ q ′ o r q ′ ≼ 0 q \preccurlyeq q’\;or\;q’ \preccurlyeq 0 , then we have that for every p , q ∈ B , p ≽ q p,q \in B,p \succcurlyeq q implies μ p ⩾ μ q \mu p \geqslant \mu q .