  We prove H\"ormander's type hypoellipticity theorem for stochastic partial
differential equations when the coefficients are only measurable with respect
to the time variable. The need for such kind of results comes from filtering
theory of partially observable diffusion processes, when even if the initial
system is autonomous, the observation process enters the coefficients of the
filtering equation and makes them time-dependent with no good control on the
smoothness of the coefficients with respect to the time variable.
