  The nonlinear filter associated with the discrete time signal-observation
model $(X_k,Y_k)$ is known to forget its initial condition as $k\to\infty$
regardless of the observation structure when the signal possesses sufficiently
strong ergodic properties. Conversely, it stands to reason that if the
observations are sufficiently informative, then the nonlinear filter should
forget its initial condition regardless of any properties of the signal. We
show that for observations of additive type $Y_k=h(X_k)+\xi_k$ with invertible
observation function $h$ (under mild regularity assumptions on $h$ and on the
distribution of the noise $\xi_k$), the filter is indeed stable in a weak sense
without any assumptions at all on the signal process. If the signal satisfies a
uniform continuity assumption, weak stability can be strengthened to stability
in total variation.
