An infinite complete
Boolean algebra satisfies |B|ℵ0= |B| (where || denotes cardinality). This is a theorem
of R. S. Pierce, derived in consequence of his general decomposition theorem [9]. It is
here shown (directly) that |B|ℵ0= |B| for B merely countably complete; this has the
corollary (actually, equivalent) that if A is an algebra of measurable functions modulo
null functions, and D is a subset of A which is dense in the uniform topology, then
|D| = |A|. The relation |B|f= |B| for f-complete Boolean algebras B is considered;
the main result is a structure theorem for the nontrivial counterexamples (which are
shown to exist abundantly).
