We have the following settings:
Given the probability space $(\Omega,\mathscr F,P)$
$T$ is a random time, if it is a $\mathscr F$-measurable random variable with values in $[0,\infty]$
A stochastic process is called measurable if for every $A\in\mathscr B(\Bbb R^d)$, the set $\{(t,\omega);X_t(\omega)\in A\}$ belongs to the product $\sigma$-algebra $\mathscr B([0,\infty))\otimes\mathscr F$. I will be appreciate for any help.
Now I want to show that for a measurable process $X_t$(with well-defined $X_{\infty}$) and a random time $T$, the collection of all sets of the form $\{X_T\in A\}$ and $\{X_T\in A\}\cup\{T=\infty\}$,forms a sub-sigma algebra of $\mathscr F$.
I understand that we have the following mapping$$\omega\rightarrow(\omega,T(\omega))\rightarrow X(\omega,T(\omega))$$ $$\mathscr F\rightarrow\mathscr B([0,\infty])\otimes\mathscr F\rightarrow\mathscr B([0,\infty])$$.
As $X_t$ is measurable, we know that $X_T^{-1}(A)$ is somehow contained in $\mathscr B([0,\infty])\otimes\mathscr F$. As I am not sure if the preimage of $X$ can be written in the form of $C\otimes D$ and then consider $T^{-1}(D)\cap C$ in the product sigma algebra, I don't know how to proceed further to the original space $\mathscr F$