Exercise 1.5.14 (Equivalent formulation of uniform integrability II)

$$ We verify the conditions from Definition 1.5.11:

(i)

Uniform bound on $L^1$ norm is satisfied by theorem assumption directly.

(ii)

To demonstrate that $\underset{M\to \infty }{\lim <!--\; FUNCTION\; APPLICATION\; -->}\underset{n\in \mathbb{N}}{\sup <!--\; FUNCTION\; APPLICATION\; -->}{\mathrm{\int \; <!--\; nolimits\; -->}<!--\; FUNCTION\; APPLICATION\; -->}_{\{|f_n|\ge M\}}\left|f_n\right|=0$ fix an arbitrary $𝜖>0$. By theorem assumption, we can find a $\delta >0$ such that the integral of $\left|f_n\right|$ on any set with measure less than $\delta$ is less than $𝜖$. By Markov’s inequality combined with the uniform bound on $L^1$ norm, the sets $\{x\in X:|f_n(x)|\ge M\}$ must decrease as $M\to \infty$: $$\forall <!--\; FUNCTION\; APPLICATION\; -->n\in \mathbb{N}:\{x\in X:|f_n(x)|\ge M\}\le \frac{1}{M}\left|f_n\right|\le \frac{1}{M}\underset{k\in \mathbb{N}}{\sup <!--\; FUNCTION\; APPLICATION\; -->}\left|f_k\right|.$$

In other words, we can choose an $M$ large enough so that $\mu (\{x\in X:|f_n(x)|\ge M\})\le \delta$. We then have

$$\forall <!--\; FUNCTION\; APPLICATION\; -->n\in \mathbb{N}:{\int \; <!--\; nolimits\; -->}_{\{|f_n|\ge M\}}\left|f_n\right|\le 𝜖.$$

(iii)

Here we use the assumption of a finite measure space / probability space. By monotonicity we have $$\underset{M\to 0}{\lim <!--\; FUNCTION\; APPLICATION\; -->}\underset{n\in \mathbb{N}}{\sup <!--\; FUNCTION\; APPLICATION\; -->}{\int \; <!--\; nolimits\; -->}_{\{|f_n|\le M\}}\left|f_n\right|d\mu \le \underset{M\to 0}{\lim <!--\; FUNCTION\; APPLICATION\; -->}\underset{n\in \mathbb{N}}{\sup <!--\; FUNCTION\; APPLICATION\; -->}P(\{\left|f_n\right|\le M\})\cdot M\le \underset{M\to 0}{\lim <!--\; FUNCTION\; APPLICATION\; -->}M=0$$

as desired.