Consider the space of RCLL (Cadlag) functions on the domain $[0,1]$ and endowed with the Skorohod topology. Let us consider the set $S := \{x: \omega_x (\delta) \leq \epsilon\}$, where $\omega_x (\cdot)$ is the modulus of continuity, defined as $\omega_x(\delta) = \sup_{s,t:|s-t|\leq \delta} |x(s)-x(t)|$.
Problem: How do we show that the set $S$ is measurable??