Showing continuity of integrals of Feller process Let $\\{P^x\\}$ be the probability distribution for a Feller process $\\{X\\}_t$. Then, how does one show that $$F(x) = \int_Df(X_t)g(X_s)P^x(d\omega)\quad (\dagger)$$ is a continuous function of $x$ given continuous, bounded functions $f$ and $g$? The Feller property immediately tells that $$G(x) = \int_Df(X_t)P^x(d\omega)$$ is continuous for such an $f$, but I do not see how to extend this to derive $(\dagger).$

Hint: assume wlog that $t>s$. By the Markov property, $$ F(x) = E^x[f(X_t) g(X_s)] = E^x[ h(X_s)g(X_s)], $$ where $h(x) = E^x[f(X_{t-s})]$.