oo, x] = tt(0) = 0 = (v o y-

Let fl11 be a Cauchy sequence in (i1'(S), d), and let G11 be the d.f. of Jtw Then supxEIH!I IG11 (x)- G111 (x)l = supxEsiG"(x)- G111 (x)l ---+ 0 as n, m---+ oo. By the completeness of ~ 1 , there exists a function G(x) such that sup,EIH!I IG,l~) - G(x)l ---+ 0. It is simple to check that G is the d.f. of a probability measure v, say, on ~ 1 • We need to check that v(S) = I. For this, given c > 0 find n" such that supxiG11 (x)- G(x)l < c/2 Vn;:::: n<. Now find Yo: E S such that G11 (yE) > l-Ej2. It follows that G(yc) > I-€. In the