Two questions about uniform integrability
Solution 1:
A family of random variables $(X_{\alpha})_{\alpha\in A}$ is u.i. if $$ \sup_{\alpha\in A}\mathsf{E}|X_{\alpha}|1\{|X_{\alpha}|>M\}\to 0 $$ as $M\to\infty$. Note that $X_{\alpha}$'s may have different distributions (densities if exist). If $(Y_{\alpha})_{\alpha\in A}$ is a u.i. family of r.v.s. satisfying $|X_{\alpha}|\le |Y_{\alpha}|$ a.s. for all $\alpha\in A$, then $(X_{\alpha})_{\alpha\in A}$ is u.i. as well because for any $\alpha\in A$, $$ \mathsf{E}|X_{\alpha}|1\{|X_{\alpha}|>M\}\le \mathsf{E}|Y_{\alpha}|1\{|Y_{\alpha}|>M\}. $$