Two questions about uniform integrability

Question

I have two questions about the uniform integrability.

The definition I am using is that a class of random variables $\mathbb{\chi}$ is uniformly integrable if given an $\epsilon >0$, there exists a $k$ such that for any $x$ in $\chi$ we have $\mathbb{E}[|x| \mathbb{I}\{x \geq k\}] < \epsilon$.

First, is that in the definition of uniform integrability, can the density function which we compute the expectation with respect to it, vary with $n$?

Second, suppose I have a sequence of random variables $(x_n)_{n\geq 1}$ (with varying density function w.r.t n). For the two functions $f,g$, I know that $f(x) \leq g(x)$ for all $x$. Does the uniform integrability of the sequence $(g(x_n))_{n \geq 1}$ imply the uniform integrability of $(f(x_n))_{n \geq 1}$?

Answer 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 $\alpha\in A$, $$ \mathsf{E}|X_{\alpha}|1\{|X_{\alpha}|>M\}\le \mathsf{E}|Y_{\alpha}|1\{|Y_{\alpha}|>M\}. $$