  When is a nonlinear filter stable with respect to its initial condition? In
spite of the recent progress, this question still lacks a complete answer in
general. Currently available results indicate that stability of the filter
depends on the signal ergodic properties and the observation process regularity
and may fail if either of the ingredients is ignored. In this note we address
the question of stability in a particular weak sense and show that the
estimates of certain functions are always stable. This is verified without
dealing directly with the filtering equation and turns to be inherited from
certain one-step predictor estimates.
