A unified treatment of filtering in stochastic distributed systems is presented using a function space approach. An infinite-dimensional analogue of the well-known matrix inversion lemma is established and used in developing explicit and computationally efficient representations for the estimate and covariance equations. It also enables us to pursue the analogy to the finite-dimensional case and certain new results are obtained for the case of intermittent measurements. The equations for the case of continuous-time measurements follow easily from the above under passage to the limit. Measurements over the entire region as well as those at a finite number of locations are considered.