Continuous time Stochastic Process stopping time measurability

Question

Let $\{X_t,\mathcal{F}_t;0\leq t < \infty\}$ be continuous time stochastic processes and $T$ be $\{\mathcal{F}_t\}_{0\leq t < \infty}$ stopping time. How to prove $X_T$ is $\mathcal{F}_T$ measurable?

Answer 1

We need additional hypothesis that $X_t$ is progressively measurable and $T$ is finite a.s-P. With this hypothesis we can prove that $X_{\min \{T,t\}}$ is progressively measurable and hence that $X_T$ is $\mathcal{F}_T$ measurable. It is a theorem in chapter 1 of the book "Stochastic Calculus and Brownian Motion" by Karatzas and Shreve.