ABSTRACT

Prp(X x R) = {p E PT(X x 0) : Top= P ) . Both Pr12(X) and Prp(X x R) carry a canonical convex structure in the sense that for any two random measures p and v and any p 6 [ O , 1 ] also pp+ (1 -p)v is a random measure, given by (pp, + (1 -p) v,) (B) = pp,(B) + (1 -p)v, (B). For Prp(X x R) the convex structure is simply given by pointwise addition. By Propositions 3.3 (ii) and 3.6 we get an isomorphism between Prn(X) and Prp(X x R), given by (3.1), or equivalently by

for all A E 9 @ 9. The convex structure is preserved by this isomorphism. We will usually identify Prp(X x R) and P m ( X ) via these relations without further noticc. Henceforth we will only speak of a random measure p E Prn(X) . It should become clear from the respective context when p is to be understood as a transition probability from R to X, and when it is to be understood as a measure on the product space X x R.