** Authors:**
EISSA D. HABIL

** 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
\sigma-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.

** 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.

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