Exchangeable sequences driven by an absolutely continuous random measure

Let $S$ be a Polish space and $(X_n:ngeq 1)$ an exchangeable sequence of $S$-valued random variables. Let $alpha_n(cdot)=Pigl(X_n+1incdotmid X_1,ldots,X_nigr)$ be the predictive measure and $alpha$ a random probability measure on $S$ such that $alpha_noversetweaklongrightarrowalpha$ a.s.. Two (related) problems are addressed. One is to give conditions for $alphalllambda$ a.s., where $lambda$ is a (non random) $sigma$-finite Borel measure on $S$. Such conditions should concern the finite dimensional distributions $mathcalL(X_1,ldots,X_n)$, $ngeq 1$, only. The other problem is to investigate whether $ ormalpha_n-alphaoverseta.s.longrightarrow 0$, where $ ormcdot$ is total variation norm. Various results are obtained. Some of them do not require exchangeability, but hold under the weaker assumption that $(X_n)$ is conditionally identically distributed, in the sense of citeBPR04.