Cardinality of k-complete Boolean algebras

Abstract

An infinite complete Boolean algebra satisfies \B\*o = \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 |£|*O = |JB| 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\* = \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).