ABSTRACT
PROOF Measurabilit,~ of w I-, p,(C(w)) and of w r-+ p,(Ch(w)) for 6 > 0 follows from Corollary 3.4 (invoking measurability of graph(C"), which follows from Proposition 2.4). Closedness of C implies almost sure convergence of p,(C6(w)) to p,(C(w)) for 6 4 0. The convergence is, iri addition, rnonotone, so L1(P)-convergence follows from tjhe monotone convergence theorern. Finally, for 6 > 0 put g6(z, w) = 1 - rnin{l, d(:c, C(w))/6}. Then
4.9 Proposition If for some topology on P%(X) (not necessarily the narrow topology) /L + p(g) is a continuous map from Prn(X) to R for every g E BLn(X) with 0 < g < 1 and [g(.,w)lL < 1 ( P - a x ) , then 11 + p( f ) is contin,uou.s in this topology for every f E Ca(X). PROOF For g E BLn(X) with 0 < g < c ant1 [g ( . ,w) ]~ < c for sorne 0 < c E R, continuity of LL + p(g) is inlmrdiate by multiplication with l/c.
