Exchangeable sequences driven by an absolutely continuous random measure

Let $S$ be a Polish space and $(X_n:n\geq 1)$ an exchangeable sequence of $S$-valued random variables. Let $\alpha_n(\cdot)=P\bigl(X_{n+1}\in\cdot\mid X_1,\ldots,X_n\bigr)$ be the predictive measure and $\alpha$ a random probability measure on $S$ such that $\alpha_n\overset{weak}\longrightarrow\alpha$ a.s.. Two (related) problems are addressed. One is to give conditions for $\alpha\ll\lambda$ a.s., where $\lambda$ is a (non random) $\sigma$-finite Borel measure on $S$. Such conditions should concern the finite dimensional distributions $\mathcal{L}(X_1,\ldots,X_n)$, $n\geq 1$, only. The other problem is to investigate whether $\norm{\alpha_n-\alpha}\overset{a.s.}\longrightarrow 0$, where $\norm{\cdot}$ 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 \cite{BPR04}.