ABSTRACT
Proof Let P11 (n ~ l) be a Cauchy sequence in (.OJ(S),d). Let P11 be the extension of P11 to ~k, i.e., P11 (B) := P11 (B n S), BE .:!6'(~k). By taking ¢ in Eq. (7.5) to be the identity map, it is seen that the distribution functions (d.f.'s) F11 of P11 are Cauchy in the uniform distance for functions on~". The limit F on IR" is the distribution function of a probability measure P on ~k, and P11 converges weakly toP on ~k. On the other hand, P11 (A) converges to some Px(A) uniformly \:1 A E sf. It follows that Px(A) = P(A) for all A = ( -oo, x] n S (\:lx E ~"). To show that P(S) = I recall that since S is closed and P11 (S) = l \:In, by Alexandroff's theorem (Billingsley, (1968) pp. 11, 12) one has
P(S) ~ lim P11 (S) = lim P11 (S) = l. /1-'>X Jl--l-'X
distribution function F of 1-L uniformlay on ~ (since, considered as probability measures on ~. J-L11 converges weakly to 1-L and F is continuous). But 1-L tj. .:!Jl(5). Thus, although Jt11 is a Cauchy sequence in the d-metric it has no limit in .:1>(5).
