Let $\left\{\mathcal R^{[0, 1]}, \overline {\mathscr B}\left(\mathcal R^{[0, 1]}\right), P\right\}$ be a probability space. If the process $\left\{\mathcal x_t, t \in [0, 1]\right\}$ is separable, then the event $$\left\{ x_t\text{ continuous at }t_0\right\}$$ is measurable. How can I start to evidence it?
I consider, that $$\left\{\mathcal x_t \text{ continuous at } t_0\right\} = \left\{\lim_{t\to t_0} x_t = x_{t_0}\right\}$$ and don't know what to do further.
By the way, let complete probability space $\{\Omega\ , F, P\}$, $\{E, \mathcal E\}$ is compact. $T$ is separable set. The process $\{\mathcal x_t ,t \in T\}$ is separable, if countable dense subset $T_0$ exists in $T$, and exists $C \subset F: P(C)=0$, that for all $F$ (closed sets) in $E$, for all $G$ (open sets) in $T$, $$\left\{x_t \in F,t \in G\right\}\; \Delta\; \left(\left\{ x_t \in F,t \in G\right\}\cap\ T_0\right) \subset C.$$
Start with $\{\lim_{t\to t_0} x_t = x_{t_0}\}\cap_{n=1}^\infty\cup_{m=1}^\infty\{|x_t-x_{t_0}|<1/n,\forall t\in(t_0-1/m,t_0+1/m)\}$. By separability, the difference between $\{|x_t-x_{t_0}|<1/n,\forall t\in(t_0-1/m,t_0+1/m)\}$ and $\{|x_t-x_{t_0}|<1/n,\forall t\in(t_0-1/m,t_0+1/m)\cap T_0\}$ is contained in the null set $C$.