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We consider a signal process X taking values in a complete, sep- arable metric space E. X is assumed to be a Markov process charachterized via the martingale problem for an operator A. In the context of the finitely additive white noise theory of filtering, we show that the optimal filter it(y) is the unique solution of the analogue of the Zakai equation for every y, not necessarily continuous. This is done by first proving uniqueness of solution to a (perturbed) measure valued evolution equation associated with A. An additional assumption of uniqueness of the local martingale problem for A is imposed.

Uniqueness of solution to the Kolmogorov forward equation: applications to white noise theory of filtering