Example of almost surely continuous stochastic process $(X_t)$ with $\{\omega \mid t\mapsto X_t(\omega )\text{ continuous}\}$ non measurable?

Question

Does someone has an example of stochastic process $(X_t)$ that is almost surely continuous but $\{\omega \mid t\mapsto X_t(\omega )\text{ continuous}\}$ is not measurable ? It look strange for me. Because if $(X_t)$ is a.s. continuous then $\{\omega \mid t\mapsto X_t(\omega )\text{ continuous}\}^c$ has measure $0$ and thus is measurable.

Answer 1

Null sets are measurables in complete measurable spaces only. If it's not complete, there are null sets that are not measurable. Take for example $\Omega =[0,1]$, $\mathcal F=\mathcal B([0,1])$ the Borel set of $[0,1]$ and $\mathbb P$ the Lebesgue measure. Let $N$ a null set that is not a Borel set (such set exist). Then $$X_t(\omega )=\begin{cases}t&\omega \notin N\\ \boldsymbol 1_{\mathbb Q\cap [0,1]}(t)&\omega \in N\end{cases},$$ is such an example. If you measure space is complete (i.e. null set are measurable), then indeed, $\{\omega \mid t\mapsto X_t(\omega )\}$ is measurable for the reason you said.