  Weak Feller property of controlled and control-free Markov chains lead to
many desirable properties. In control-free setups this leads to the existence
of invariant probability measures for compact spaces and applicability of
numerical approximation methods. For controlled setups, this leads to existence
and approximation results for optimal control policies. We know from stochastic
control theory that partially observed systems can be converted to fully
observed systems by replacing the original state space with a probability
measure-valued state space, with the corresponding kernel acting on probability
measures known as the non-linear filter (belief) process. Establishing
sufficient conditions for the weak Feller property for such processes is a
significant problem, studied under various assumptions and setups in the
literature. In this paper, we prove the weak Feller property of the non-linear
filter process (i) first under weak continuity of the transition probability of
controlled Markov chain and total variation continuity of its observation
channel, and then, (ii) under total variation continuity of the transition
probability of controlled Markov chain. The former result (i) has first
appeared in Feinberg et. al. [Math. Oper. Res. 41(2) (2016) 656-681]. Here, we
present a concise and easy to follow alternative proof for this existing
result. The latter result (ii) establishes weak Feller property of non-linear
filter process under conditions, which have not been previously reported in the
literature.
