An extension of Skorohod’s almost sure representation theorem

Abstract:

Skorohod discovered that if a sequence ${Q_n}$ of countably additive probabilities on a Polish space converges in the weak star topology, then, on a standard probability space, there are ${Q_n}$-distributed ${f_n}$ which converge almost surely. This note strengthens Skorohod’s result by associating, with each probability $Q$ on a Polish space, a random variable ${f_Q}$ on a fixed standard probability space so that for each $Q$, (a) ${f_Q}$ has distribution $Q$ and (b) with probability 1, ${f_P}$ is continuous at $P = Q$.