I am doing a question Let $X$ be a continuous local martingale and suppose $\mathbb{E}\left[\sup\limits_{0\leq s\leq t} |X_s|\right]<\infty$ for each $t\geq 0$. Then $X$ is a true martingale.
In the solution, it says the following- since $|X_{\min(T,t)}|\leq \sup\limits_{0\leq s\leq t} |X_s|$ we can conclude that $\{X_{\min(T,t)} \mid T\}$ is uniformly integrable.
I am not sure how he got this step. Hasn't he only shown $X_{\min(T,t)}$ is $L_1$ bounded? This doesn't imply uniform integrability.
The family $\{X_{\min\{T,t\}},t\geqslant 0, T\mbox{ stopping time }\}$ is more than bounded in $\mathbb L^1$: it is uniformly bounded (in $t$ and $T$ by an integrable random variable). If $\mathcal F:=\{f_i,i\in I\}$ is a family of functions such that $\sup_{i\in I}|f_i|$ is integrable, then $\mathcal F$ is uniformly integrable.