A Radon-nikodym Theorem for Stone Algebra Valued Measures

Introduction. Let C(S) be the ring of continuous real valued functions on a compact Hausdorff space S. Stone [5] shows that each bounded subset of C(S) has a least upper bound (in C(S)) if and only if the closure of each open subset of S is open ; in this event we call C(S) a Stone algebra. Throughout this paper C(S) is a Stone algebra. It is convenient to adjoin an object +00, not in C(S), and extend the partial ordering of C(S) to C(S) u { + 00} in the obvious way. When {an : n = 1, 2,...} is an unbounded set in C(S) we define V"=i on to be +00. Following [6] we define a C(<S')-valued measure on a measurable space (X, 01) to be a map m:0i-+ C(S) u { + 00} such that mE^O for each E and, if {En} «=1,2,... is a pairwise disjoint sequence of sets in 0) then