By Cihan Mert, Mehmet Özüm

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C„>£) < e/3». _\ ^ . - LetC = {JnCn_,. Since μ is a measure we can choose a & so large that μ{ϋ — JJ» = i Q e) < e/2. · εΙ2<Σ^1εβ'> + εΙ2 = ε. The two preceding arguments show that & is a σ-algebra. To complete the proof it is enough to show that & contains all closed sets. Let C C X be a closed set and ε > 0. , U1 2 i/ 2 2 . . such that C = Π ^ ° = 1 I7n. Since μ(ϋη) -+ μ{0) we can find an n0 such that μ(υηο — C) < ε. If we define Ce = C, Ue = £/Wo we see that μ(£/6 — Ce) < ε. This completes the proof.

Then X is dense in Xv Any g e U(X) can be extended uniquely to a gGC(X1). Further, s u p ^ ^ \g{x)\ = su x n Pxex \i( )\I other words, the Banach spaces U(X) and C(X1) are isomorphic. Since Xx is a compact space, C(XX) is separable. This shows that U(X) is separable. T h e o r e m 6*2 Jt(X) can be metrized as a separable metric space if and only if X is a separable metric space. Suppose X is a separable metric space. Then by the celebrated theorem of Urysohn (cf. Kelley [16], p. 125), X can be topologically imbedded in a countable product of unit intervals.

Let A be a nonnegative linear functional on C(X) such that Λ(1) = 1. Then there exists a unique finitely additive regular measure μ on s/x (the algebra generated by all the open subsets of X) such that A{f) = ydM, feC(X) Conversely, if μ is a finitely additive measure on s/x f -►/ /' άμ is nonnegative, linear, and Λ(\) = 1. then the map Λ: 36 II. PROBABILITY MEASURES IN A METRIC SPACE PROOF. 3. Let / be any function in C(X) such that 0 < / < 1. We shall first establish that A{f ( > ίχ f άμ. To this end let n be any integer and let Gi — {x\ f(x) > ijn).