This paper presents a generalization of results on convergence and robustness of discretization schemes for nonlinear filtering proposed by Kushner. This is made possible by a general theorem on the convergence of semigroups of operators on a Banach space, which gives sufficient conditions for a semidiscretization scheme to remain convergent, once the time is implicitly discretized. As a consequence, sufficient conditions can be given for selecting space discretizations of the state process generator to construct computable nonlinear filters converging to the optimal one.