ABSTRACT
In order to illustrate the differences between the topologies we take the simplest nontrivial Polish space X , which is the two point set X = {a, b), a # b, with the discrete topology. (It makes the presentation clearer not to choose a = 0 and b = 1.) All function spaces (measurable, continuous, Lipschitz) coincide and are equal to RX, which can be identified with R2. We will denote functions f : {a, b) x R + R by f = (fa, fb ) with fa, f b : R + R, where fa(w) = f (a, w) and fb(w) = f (b , w). The space Cn(X) of random continuous functions is thus given by collecting all f : R -+ R2 with w * max{l fa(w)l, I fb(w)I) integrable with respect to P. Since all norms on R2 are equivalent, Cs2 ( X ) can be identified with L1 (R, 9, P; R2). Denote by I . I an arbitrary norm on R2. The space BLn(X) of random bounded Lipschitz functions is given by g : fl + R2 with w ~?r Ig(w)l bounded P-a.s., so BLn(X) can be identified with Lm(R, 9, P ; Kt2). The space Pr (X) of probability measures consists of the Bernoulli measures, P r ( X ) = {tb, + (1 - t)bb : t E [0,1]). In order to be specific, choose d(p, <) = sup{p(g)-((g) : g E BL(X), 0 5 y < 1) for a metric on P r ( X ) , metrising the narrow topology. In the present situation this gives for p = tb, + (1 - t )& and < = sh, + (1 - s)hb with t , s E [ O , 1 ] , and !?a = g(a), .9b = g(b),
Any random measure LL E Prn(X) is given by
where w H t ( w ) is measurable (since w ++ pu({a)) is measurable). Thus Prn(X) can be identified with L = {t : R -t [0,1] : t measurable), where the identification is given by (w ++ ,u,) H (w H p,({a)) = t(w)). The set of Young rrleasures consists of the Dirac measures bv(,) = t(w)S, + (1 - t(w))bb,
given by random variables y : R + {a, b}, or equivalently by t : R -+ (0, I ) , making up the set {t : R + {0,1) : t measurable) C L. We then have L c Lp(R, 9, P)for every 1 < p < m. For p = co, L is homeomorphic to the closed unit ball in L" ( R , 9 , P) (with horneornorphism given by f * 2 f - 1). In particular, L has nonempty interior with respect to the Lm-topology. This is not the case for p < m. From (5.3) we have
with ,LL and u represented by t and s , respectively.
