A Stone’s Representation Theorem and Some Applications
نویسنده
چکیده
In this paper, we prove the following form of Stone’s representation theorem: Let ∑ be a σ -algebra of subsets of a set X . Then there exists a totally disconnected compact Hausdorff space K for which ( ∑ ,∪,∩) and (C(K),∪,∩) , where C(K) denotes the set of all clopen subsets of K , are isomorphic as Boolean algebras. Furthermore, by defining appropriate joins and meets of countable families in C(K) , we show that such an isomorphism preserves σ -completeness. Then, as a consequence of this result, we obtain the result that if ba(X, ∑ ) (respectively, ca(X, ∑ )) denotes the Banach space (under the variation norm) of all bounded, finitely additive (respectively, all countably additive) complex-valued set functions on (X, ∑ ), then ca(X, ∑ )=ba(X, ∑ ) if and only if (1) C(K) is σ -complete; and if and only if (2) ∑ is finite. We also give another application of these results. AMS Subject Classification (1991). Primary: 28A33, 28B05; Secondary: 06 C15, 03G12, 81P10.
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تاریخ انتشار 2002