Optimal Stopping Theorem for UI Martingales

Question

In my notes, I was given that:

"Let $X_n$ be an $F_n$ (super,sub) martingale and $\tau$ an $F_n$ stopping time. Then $X_{n \wedge \tau}$ is a (super, sub) martingale wrt $F_n$ and $F_{n \wedge \tau}$"

However in the section about Uniformly Integrable Martingales it says:

"Let $X_n$ be a UI Martingale. Then for any $F_n$ stopping time $\tau$, $X_{n \wedge \tau}$ is Uniformly integrable."

Is it not true that if $X_n$ is a UI Martingale then $X_{n \wedge \tau}$ is a Uniformly integrable martingale?

Answer 1

If $\{X_n\}$ is UI then $X_n=E(X|\mathcal F_n)$ for some integrable $X$ and $X_{n\wedge \tau }=E(X|\mathcal F_{n\wedge \tau })$ which makes $\{X_{n\wedge \tau\} }\}$ also UI.