A Stone`s Representation Theorem and Some Applications

Authors : Eissa D. HABIL

Pages : 287-299

View : 9 | Download : 6

Publication Date : 0000-00-00

Article Type : Research Paper

Abstract :In this paper, we prove the following form of Stone`s representation theorem: Let \sum be a s-algebra of subsets of a set X. Then there exists a totally disconnected compact Hausdorff space {\cal K} for which insert ignore into journalissuearticles values(\sum, \cup, \cap); and insert ignore into journalissuearticles values({\cal C}insert ignore into journalissuearticles values({\cal K});, \cup ,\cap);, where {\cal C}insert ignore into journalissuearticles values({\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}insert ignore into journalissuearticles values({\cal K});, we show that such an isomorphism preserves s-completeness. Then, as a consequence of this result, we obtain the result that if bainsert ignore into journalissuearticles values(X,\sum); insert ignore into journalissuearticles values(respectively, cainsert ignore into journalissuearticles values(X,\sum);); denotes the Banach space insert ignore into journalissuearticles values(under the variation norm); of all bounded, finitely additive insert ignore into journalissuearticles values(respectively, all countably additive); complex-valued set functions on insert ignore into journalissuearticles values(X, \sum);, then cainsert ignore into journalissuearticles values(X, \sum);=bainsert ignore into journalissuearticles values(X, \sum); if and only if insert ignore into journalissuearticles values(1); {\cal C}insert ignore into journalissuearticles values({\cal K}); is s-complete; and if and only if insert ignore into journalissuearticles values(2); \sum is finite. We also give another application of these results.
Keywords : Boolean ring, Boolean space, Stone space, Stone representation, bounded finitely additive set function, countably additive set function, convergence of sequences of measures, weak topology