Non uniformly integrable stochastic process if we consider a larger range This is probably a stupid question, but am I mislead if I think that as soon as a stochastic process indexed by $t$ (continuous time) is not uniformly integrable (UI) for a certain range of $t$, say on $[0,1]$, it is also not UI on $[0,\infty]$? And if that is wrong, then why so? Thank you in advance!

For any collections $C$ and $D$ of random variables, if $D$ is uniformly integrable and $C\subseteq D$, then $C$ is uniformly integrable.

To see this, recall that $D$ is uniformly integrable if and only if $S(x,D)\to0$ when $x\to+\infty$, where $$ S(x,D)=\sup\\{\mathrm E(|X| ; |X|\geqslant x)\mid X\in D\\}. $$ Now, if $C\subseteq D$, then $S(x,C)\leqslant S(x,D)$ for every $x$, hence the conclusion follows.

Thus, the contraposition holds: if $C$ is not uniformly integrable and $C\subseteq D$, then $D$ is not uniformly integrable.