Suppose that $X$ have finite measure, let $1<p<\infty$, and suppose that $f_n : X\rightarrow \mathbb{R}$ is a sequence of measurable functions such that $\sup_n \int_X |f_n|^p d\mu < \infty$. Show that the sequence $f_n$ is uniformly integrable. In another word, show that
$$\sup_n \int_X |f_n|^p d\mu < \infty \;\;\;\Longrightarrow\;\;\; \sup_n \int_{|f_n|>M} |f_n| d\mu \rightarrow 0 \text{ when } M\rightarrow \infty.$$
I have tried using contradiction, but I am not sure how to use the power $p$ in the problem.
I can see why it would work, in some sense, the power $p$ gets rid of the case often called "escape to vertical infinity". For example, define $f_n :[0,1] \rightarrow \mathbb{R}$ with $f_n = n\chi_{[o,\frac{1}{n}]} $, without the power $p$, we have
$$\sup_n \int_X |f_n| d\mu =1 \;\;\text{ and }\;\; \sup_n \int_{|f_n|>M} |f_n| d\mu =1 \text{ for each } M,$$
but
$$\sup_n \int_X |f_n|^p d\mu = \infty$$
Best Answer
Use the inequality $$\chi\{|f|>R\}\cdot |f(x)|\cdot R^{p-1}\leqslant |f(x)|^p,$$ where $\chi(A)$ denotes the indicator function of the set $A$. Then integrate to obtain $$\int_{\{|f_n|>R\}}|f_n|\mathrm d\mu(x)\leqslant R^{1-p}\sup_k\int_X\left|f_k(x)\right|^p\mathrm d\mu(x).$$