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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 $$\sup_n \int_X |f_n|^p d\mu = \infty$$

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    $\begingroup$ Use Hölder. ($\int_E |f|\le\Vert f\Vert_p\Vert \chi_E\Vert_q$.) $\endgroup$ Mar 27, 2014 at 17:19
  • $\begingroup$ @DavidMitra What "particularization" of Hölder is this? $\endgroup$
    – mavavilj
    Oct 10, 2019 at 20:39

1 Answer 1

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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 $$\int_{\{|f_n|>R\}}|f_n|\mathrm d\mu(x)\leqslant R^{1-p}\sup_k\int_X\left|f_k(x)\right|^p\mathrm d\mu(x).$$

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  • $\begingroup$ Yes, thank you. I used the hint given above $\int_{\{f\geq M\}} f \leq ||f||^p ||\chi_{\{f\geq M\}}||^q$, where $||f||^p$ is finite, and $||\chi_{\{f\geq M\}}||^q$ goes to zero as $m\rightarrow \infty$ $\endgroup$
    – Xiao
    Apr 3, 2014 at 11:00
  • $\begingroup$ Does that inequality have a name? $\endgroup$
    – vlovero
    Apr 16, 2021 at 20:27

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