\displaystyle  \begin{aligned} & \int f(U^{x,n}_t)d\mu(x)\rightarrow \int f(U^{x}_t)d\mu(x),\\ & \int\xi^{x,n}_sf^\prime(U^{x,n}_{s-})d\mu(x)\rightarrow\int\xi^{x}_sf^\prime(U^{x}_{s-})d\mu(x) \end{aligned} 