On the existence of Markovian measures on continuous paths

Let $\mathbf{\eta }$ be a positive Radon measure on the space of continuous paths from R into a locally compact Polish space $Y$, and assume that $\mathbf{\eta }$ admits an invariant measure. We explicit conditions on $\mathbf{\eta }$ such that successive Markovianisation of $\mathbf{\eta }$ over any dense and countable set of times have all limit points (in the weak-star topology on Radon measures) satisfying the strong Markov property. We show that if Y is a locally compact Polish group, and $\mathbf{\eta }$ is left or right translation invariant, then $\mathbf{\eta }$ satisfies such conditions. Our proof uses the Zermalo-Fraenkel and Axiom of Dependant Choice axiomatisation of set theory, in which countable products of compacts are compact.