Let $(X_{n})_{n\in\mathbb{N}}$ be $(\mathcal{F}_{n})_{n\in\mathbb{N}}$-adapted and let $T$ be a $(\mathcal{F}_{n})_{n\in\mathbb{N}}$-stopping time. Then $$X_{T}\text{ is a Random Variable}$$and $$X_{T}\text{ is }\mathcal{F}_{T}\text{-measurable}$$

Let $(X_{t})_{t\ge 0}$ be $(\mathcal{F}_{t})_{t\ge 0}$-adapted and let $T$ be a $(\mathcal{F}_{t})_{t\ge 0}$-stopping time. Assume $(X_{t})_{t\ge 0}$ is right continuous (respectively left) then $$X_{T}\mathbb{1}_{\{ T<\infty \}}\text{ is }\mathcal{F}_{T}\text{-measurable}$$where $$X_{T}\mathbb{1}_{\{ T<\infty \}}=\begin{cases} 0 & T(\omega)=+\infty \\ X_{T(\omega)}(\omega) & T(\omega)<\infty \end{cases}$$or if $T$ is bounded then $$X_{T}\text{ is }\mathcal{F}_{T}\text{-measurable}$$