Exercise 1.5.12 (Uniform $L^p$ bound on finite measure implies uniform integrability)

Question

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ewcommand{\intx}[1]{\int_{X} #1 \mathop{}\!\mathrm{d}{\mu}}$
Let $X$ have a finite measure, let $1<p<\infty$, and suppose that $\left(fight)_{n\in\mathbb{N}}:X	o\mathbb{C}$ is a sequence of measurable functions such that $\sup_{n\in\mathbb{N}} \intx{|f|^p}<\infty$. Show that the sequence $\left(fight)_{n\in\mathbb{N}}$ is uniformly integrable.

Elu Thingol · Accepted Answer

By \href{./exercise_1-5-11}{Exercise 1.5.11} it suffices to demonstrate that
$$\lim_{M	o\infty} \sup_{n \in \mathbb{N}} \intx{|f_n|\cdot 1_{\{|f_n| \geq M\}}} = 0.$$
Notice the relationship
$$|f_n|\cdot M^{p-1} \cdot 1_{\{|f_n| \geq M\}} \leq |f_n| ^ p \cdot 1_{\{|f_n| \geq M\}}.$$
Integrating both parts, for all $n\in\mathbb{N}$ we obtain:
$$\intx{|f_n| \cdot 1_{\{|f_n| \geq M\}}} \leq \frac{1}{M^{p-1}} \intx{|f_n| ^ p \cdot 1_{\{|f_n| \geq M\}}}.$$
The latter integral is finite, so taking limits and supremums as $M,n	o\infty$ yields zero.

Exercise 1.5.12 (Uniform $L^p$ bound on finite measure implies uniform integrability)

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Exercise 1.5.12 (Uniform $L^p$ bound on finite measure implies uniform integrability)

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