Consider a fixed probability space $(\Omega, \mathcal{F}, P)$, and suppose that the sequence of random variables $\{X_n:n\in\mathbb{N}\}$ is a discrete-time martingale with respect to some filtration $\{\mathcal{F}_n:n\in\mathbb{N}\}$. In Doob's Optional Stopping Theorem, we deal with the following objects: $X_T$ and $X_{T\wedge n}$, defined by $$\omega \mapsto X_{T(\omega)}(\omega)\qquad \text{and}\qquad \omega \mapsto X_{T\wedge n(\omega)}(\omega),$$ respectively, where $T:\Omega\rightarrow\mathbb{N}$ is a stopping time with respect to the filtration $\{\mathcal{F}_n:n\in\mathbb{N}\}$. These objects are random variables. For example, for any Borel set $B\in\mathcal{B}_{\mathbb{R}}$, it's readily seen that
\begin{align*} X_T^{-1}(B) &= \{\omega: X_{T(\omega)}(\omega)\in B\}\\ &=\bigcup_{n=1}^{\infty}\{\omega: X_n(\omega)\in B,\, T(\omega) = n\}\\ &= \bigcup_{n=1}^{\infty} \big[\{X_n\in B\}\cap \{T=n\}\big]\in\mathcal{F}, \end{align*} as $\sigma$-algebras are closed under finite intersections and countable unions.
My question deals with continuous time martingales. That is, consider a continuous time martingale $\{X_t:t\ge 0\}$ with respect to some filtration some filtration $\{\mathcal{F}_t:t\ge 0\}$. Again, Doob's Optional Stopping Theorem deals with the following objects: $X_{T\wedge t}$ and $X_{T}$, where $T\ge 0$ is a (continuous) stopping time with respect to the filtration $\{\mathcal{F}_t:t\ge 0\}$.
How does one prove that these are in fact random variables in this continuous time setting? That is, how does one prove that for any $B\in\mathcal{B}_{\mathbb{R}}$, $X_{T\wedge t}^{-1}(B)\in\mathcal{F}$ and $X_{T}^{-1}(B)\in\mathcal{F}$? One cannot simply copy the above proof because time is now uncountable. Can one discretize the positive real line in some useful way? Or perhaps another technique or trick?
Thanks for the help!