stochastic process with random variable as time, measurable

Question

i have a problem with the following exercise: Let $X$ be an measurable $\mathbb{R}^d$ valued stochastic process on $(\Omega,\mathcal{F},\mathbb{P})$ and $T$ a finite $T : \Omega\to \big[ 0, +\infty \big)$ random time. . Show that $X_T$ is $\mathcal{F}$-measurable.

I know how to show $\{\omega\in \Omega|{X_{T(\omega)}(\omega)}\in B\} \in \mathcal{F}$, but only if T would take values in $\mathbb{N}$. I have no clue how to show it for a real valued random time.

thank you

edit:Yes $X$ is measurable on the space $(\Omega \times [ 0, +\infty),\mathcal{F}\otimes \mathcal{B}([0,\infty))$

Answer 1

Thank you Bryon Schmuland, the answer of "The strong Markov property with an uncountable index set" does answer my question too. I was first put off why $T$ should be finite, but since then $\{T<\infty \}=\Omega, X_T$ is well defined for every $\omega$ without having to look at the Limits.