If a family $\{f_n\}$, is uniformly integrable, then also $\{|f_n|\}$ is.

Question

I want to show that if a set of functions $\{f_n\}_n$ is uniformly integrable, then also $\{|f_n|\}_n$ is also uniformly integrable. How can I show this? My guess is to use separate $f_n$ into real and imaginary part and then use negative and positive part for each R/I part.

Answer 1

In the following characterization of the uniform integrability of $f_n$: $$ \lim_{c\to\infty}\sup_n\int_{\{x:|f_n(x)| > c\}} |f_n(u)|du = 0 $$ it follows immediately that $(f_n)$ is uniformly integrable iff $(|f_n)|)$ is.