} \lim_{\varepsilon \rightarrow 0} \left[ \sup_{u \in B} P_u \left( \sup_{0 \leq t \leq \tau_\varepsilon - 0R_\varepsilon}\left| \frac{0}{R_\varepsilon}\int_t^{t+R_\varepsilon} f(U^{\varepsilon}(s,\cdot))ds - f(\mathbf{0})\right| > \delta \right) \right] = 0$$ for any bounded continuous function $f: C_D([0,0]) \rightarrow {{\mathbb R}}$.  Th