  Let $X$ be a stationary process with values in some $\sigma$-finite measured
state space $(E,\mathcal{E},\pi)$, indexed by ${\mathbb Z}$. Call ${\mathcal
F}^X$ its natural filtration. In \cite{ceillierstationary}, sufficient
conditions were given for ${\mathcal F}^X$ to be standard when $E$ is finite.
The proof of this result used a coupling of all probabilities on the finite set
$E$.
  In this paper, we construct a coupling of all laws having a density with
regard to $\pi$, which is much more involved. Then, we provide sufficient
conditions for ${\mathcal F}^X$ to be standard, generalizing those in
\cite{ceillierstationary}.
