  In deterministic and stochastic control theory, relaxed or randomized control
policies allow for versatile mathematical analysis (on continuity, compactness,
convexity and approximations) to be applicable with no artificial restrictions
on the classes of control policies considered, leading to very general
existence results on optimal measurable policies under various setups and
information structures. On relaxed controls, two studied topologies are the
Young and Borkar (weak$^*$) topologies on spaces of functions from a
state/measurement space to the space of probability measures on control action
spaces; the former via a weak convergence topology on probability measures on a
product space with a fixed marginal on the input (state) space, and the latter
via a weak$^*$ topology on randomized policies viewed as maps from
states/measurements to the space of signed measures with bounded variation. We
establish implication and equivalence conditions between the Young and Borkar
topologies on control policies. We then show that, under some conditions, for a
controlled Markov chain with standard Borel spaces the invariant measure is
weakly continuous on the space of stationary control policies defined by either
of these topologies. An implication is near optimality of quantized stationary
policies in state and actions or continuous stationary and deterministic
policies for average cost control under two sets of continuity conditions (with
either weak continuity in the state-action pair or strong continuity in the
action for each state) on transition kernels.
