Let $D([0,T];R^d)$ be the space of cadlag functions endowed with the usual Skorohod topology. $X_t(\omega):=\omega(t)$ denotes the usual canonical process. Assume that a family of probability measure $\mu^n$ on $D([0,T];R^d)$ is tight with a weak limit $\mu$. Then, is it true that for any bounded and continuous function $f$, we have $$ \lim_{n\to\infty}E^{\mu^n}\left(\int_0^Tf(X_r)dr\right)=E^{\mu}\left(\int_0^Tf(X_r)dr\right) ?? $$ Or is there any references for this? Thanks a lot.

Let $D([0,T];R^d)$ be the space of càdlàg functions endowed with the usual Skorohod topology. $X_t(\omega):=\omega(t)$ denotes the usual canonical process. Assume that a family of probability measures $\mu^n$ on $D([0,T];R^d)$ is tight with a weak limit $\mu$.

Then, is it true that for any bounded continuous function $f$, we have $$ \lim_{n\to\infty}E^{\mu^n}\left(\int_0^Tf(X_r)dr\right)=E^{\mu}\left(\int_0^Tf(X_r)dr\right) ? $$ Or are there any references for this? Thanks a lot.