Show that if $f_n \leq g$ for all $n$ and $g$ is integrable, then $\{f_n\}$ is uniformly integrable

Question

A sequence {$f_n$} of measurable functions is called uniformly integrable if

$$\lim_{M \to \infty} \sup_{n} \int_{[|f| >= M]} |f_n|\ \mathrm{d}\mu = 0$$

Show that if $|f_n| \leq g$ for all $n$ and $g$ is integrable, then $\{f_n\}$ is uniformly integrable.

Answer 1

Hint: $|f_n|\chi_{\{|f_n|\geqslant M\}}\leqslant g\chi_{\{g\geqslant M\}}$, so the problem reduces to show that $\lim_{M\to \infty}\int g\chi_{\{g\geqslant M\}}=0$. Use monotone convergence theorem.

The first claim of the hint follows from $\{|f_n|\geqslant M\}\subset \{g\geqslant M\}$. We can show, approximating $g$ by a simple functions, that for each $\varepsilon >0$, we can find $\delta>0$ such that for all measurable $A$ satisfying $\mu(A)<\delta$, then $\int_Agd\mu<\varepsilon$. To conclude, we have that $\mu\{g\geqslant M\}\leqslant \frac 1M\int gd\mu$.