Algebraic characterizations of measure algebras

We present necessary and sufficient conditions for the existence of a countably additive measure on a Boolean $\sigma$-algebra. For instance, a Boolean $\sigma$-algebra $B$ is a measure algebra if and only if $B-\{\boldsymbol{0}\}$ is the union of a chain of sets $C_1\subset C_2\subset ...$ such that for every $n$,

(i)

every antichain in $C_n$ has at most $K(n)$ elements (for some integer $K(n)$),

(ii)

if $\{a_n\}_n$ is a sequence with $a_n \notin C_n$ for each $n$, then $\lim_n a_n =\boldsymbol{0}$, and

(iii)

for every $k$, if $\{a_n\}_n$ is a sequence with $\lim_n a_n =\myzero$, then for eventually all $n$, $a_n \notin C_k$.

The chain $\{C_n\}$ is essentially unique.