  We consider a bivariate stationary Markov chain $(X_n,Y_n)_{n\ge0}$ in a
Polish state space, where only the process $(Y_n)_{n\ge0}$ is presumed to be
observable. The goal of this paper is to investigate the ergodic theory and
stability properties of the measure-valued process $(\Pi_n)_{n\ge0}$, where
$\Pi_n$ is the conditional distribution of $X_n$ given $Y_0,...,Y_n$. We show
that the ergodic and stability properties of $(\Pi_n)_{n\ge0}$ are inherited
from the ergodicity of the unobserved process $(X_n)_{n\ge0}$ provided that the
Markov chain $(X_n,Y_n)_{n\ge0}$ is nondegenerate, that is, its transition
kernel is equivalent to the product of independent transition kernels. Our main
results generalize, subsume and in some cases correct previous results on the
ergodic theory of nonlinear filters.
