We consider the problem of probabilistic safety verification and controller synthesis for linear time-invariant (LTI) systems with noisy state measurements. Almost no numerical results are available for safety verification of partially observable systems. We model the problem as an equivalent optimal control problem over a belief state that is a modified conditional probability density of the current state of the system. The belief state is shown to be a truncated Gaussian density in the case of LTI systems with Gaussian measurement noise, and a novel algorithm is proposed that extends existing point-based solvers to include the truncated Gaussian belief state, and continuous observation space that is adaptively gridded to reduce estimation error and increase speed of computation. Preliminary results show the method to be promising in terms of computation time as compared to other approaches.