Let $\mathcal{T} \subset \mathbb{R}$ and $\{X(t)\}_{t \in \mathcal{T}}$ be a real valued stochastic process.
Why is the following function then measurable: $$\liminf_{s \downarrow t} X(s):= \lim_{\epsilon \to 0}\; (\inf\{X(s)| s \in \mathcal{T} \cap (t,t+\epsilon)\})?$$
(We have the infimum over more then countable many measurable functions here)