The Koopmanization embeds the bilinearization via the action of the infinitesimal stochastic Koopman operator on the observables associated with the controlled nonlinear Itô stochastic differential system without explicit linearizations. The stochastic evolutions of controlled Markov processes assume the structure of controlled nonlinear Itô stochastic differential equations. This paper sketches a Koopman operator framework for the filtering of the controlled nonlinear Itô stochastic differential system. The major ingredients of this paper are the construction of the eigenfunctions, action of the infinitesimal stochastic Koopman operator, multi-dimensional Itô differential rule and filtering concerning the controlled nonlinear Itô stochastic differential system. In this paper, we illustrate the filtering in the Koopman setting for a polynomial system and compare with the filtering in the Carleman setting.