We consider a discrete time hidden Markov model where the signal is a stationary Markov chain. When conditioned on the observations, the signal is a Markov chain with stationary transition probabilities under the conditional measure. It is shown that this conditional signal is weakly ergodic when the signal is weakly ergodic and the observations are nondegenerate. This permits a delicate exchange of the intersection and supremum of sigma-fields, which has direct implications for the stability of nonlinear filters. The proof relies on an extension of results on the weak ergodicity of Markov chains in random environments to general state spaces. Finally it is shown that the main results can be lifted to the continuous time setting. The results partially resolve a long-standing gap in the proof of a result of H. Kunita (1971).