Algebraic characterizations of measure algebras

We present necessary and sufficient conditions for the existence of a countably additive measure on a Boolean σ-algebra. For instance, a Boolean σ-algebra B is a measure algebra if and only if B - {0} is the union of a chain of sets C 1 C C 2 C... such that for every n, (i) every antichain in C n has at most K(n) elements (for some integer K(n)), (ii) if {an}n is a sequence with an ∉ C n for each n, then lim n an = 0, and (iii) for every k, if {a n } n is a sequence with lim n an = 0, then for eventually all n, an∉ C k . The chain {Cn} is essentially unique.