ON THE REPRESENTATION OF <r-COMPLETE BOOLEAN ALGEBRAS

A <r-complete Boolean algebra is a Boolean algebra in which for every sequence of elements a$-, i = l, • • • , there is an element U?an, the countable union of the a», such that aiQU?an for every i, and such that if diQx for every i then U?anQx. The dual operation, countable intersection, can be introduced through complementation, and the distributive law afMJi*'a» = Uf {aC\an) and its dual can be proved (see [3, p. 93]). Certain types of Boolean algebras have representations as algebras of point sets, the representation preserving all the operations of the algebra. Among these are ordinary Boolean algebras with no further operations (Stone [l, p. 106]) and complete Boolean algebras for which very general operations and distributive laws are assumed (Tarski [2, pp. 197-198]). On the other hand it is well known that a cr-complete Boolean algebra has in general no such representation. For example., the quotient of the algebra of Lebesgue measurable subsets of [0, l ] modulo the ideal of sets of measure zero is a or-complete Boolean algebra which is not cr-isomorphic to any cr-complete Boolean algebra of point sets. The following theorem, which we shall prove in this note, shows that this example illustrates the general situation.