A Stone's representation theorem and some applications

Abstract

In this paper, we prove the following form of Stone's representation theorem: Let $sum$ be a $sigma$-algebra of subsets of a set X. Then there exists a totally disconnected compact Hausdorff space ${cal K}$ for which ($sum$, $cup$, $cap$) and (${cal C}({cal K})$, $cup$ ,$cap$), where ${cal C}({cal K}$) denotes the set of all clopen subsets of ${cal K}$, are isomorphic as Boolean algebras. Furthermore, by defining appropriate joins and meets of countable families in ${cal C}({cal K})$, we show that such an isomorphism preserves s-completeness. Then, as a consequence of this result, we obtain the result that if ba(X,$sum$) (respectively, ca(X,$sum$)) denotes the Banach space (under the variation norm) of all bounded, finitely additive (respectively, all countably additive) complex-valued set functions on (X, $sum$), then ca(X, $sum$)=ba(X, $sum$) if and only if (1) ${cal C}({cal K}$) is $sigma$-complete; and if and only if (2) $sum$ is finite. We also give another application of these results.