equivalent conditions for uniform integrability
Let $(\mathrm{\Omega},\mathcal{F},\mu )$ be a measure space^{} and $S$ be a bounded^{} subset of ${L}^{1}(\mathrm{\Omega},\mathcal{F},\mu )$. That is, $\int f\mathit{d}\mu $ is bounded over $f\in S$. Then, the following are equivalent^{}.

1.
For every $\u03f5>0$ there is a $\delta >0$ so that
$$ for all $f\in S$ and $$.

2.
For every $\u03f5>0$ there is a $K>0$ satisfying
$$ for all $f\in S$.

3.
There is a measurable function^{} $\mathrm{\Phi}:\mathbb{R}\to [0,\mathrm{\infty})$ such that $\mathrm{\Phi}(x)/x\to \mathrm{\infty}$ as $x\to \mathrm{\infty}$ and
$$\int \mathrm{\Phi}(f)\mathit{d}\mu $$ is bounded over all $f\in S$. Moreover, the function $\mathrm{\Phi}$ can always be chosen to be symmetric^{} and convex.
So, for bounded subsets of ${L}^{1}$, either of the above properties can be used to define uniform integrability. Conversely, when the measure space is finite, then conditions (2) and (3) are easily shown to imply that $S$ is bounded in ${L}^{1}$.
To show the equivalence of these statements, let us suppose that $$ for $f\in S$.
For $\u03f5>0$, property (1) gives a $\delta >0$ so that $$ whenever $f\in S$ and $$. Choosing $K>L/\delta $, Markov’s inequality^{} gives
$$ 
and, therefore, $$.
For each $n=1,2,\mathrm{\dots}$, property (2) gives a ${K}_{n}$ satisfying
$$\int {(f{K}_{n})}_{+}\mathit{d}\mu \le {\int}_{f>{K}_{n}}f\mathit{d}\mu \le {2}^{n}.$$ 
Without loss of generality, the ${K}_{n}$ can be chosen to be increasing to infinity^{}, so we can define $\mathrm{\Phi}(x)={\sum}_{n}{(x{K}_{n})}_{+}$. Then,
$$\int \mathrm{\Phi}(f)\mathit{d}\mu =\sum _{n}\int {(f{K}_{n})}_{+}\mathit{d}\mu \le \sum _{n}{2}^{n}=1.$$ 
First, suppose that $$ for $f\in S$. For $\u03f5>0$, the condition that $\mathrm{\Phi}(x)/x\to \mathrm{\infty}$ as $x\to \mathrm{\infty}$ gives a $K>0$ such that $\mathrm{\Phi}(x)/x\ge 2M/\u03f5$ whenever $x>K$. Setting $\delta =\u03f5/2K$,
$$ 
whenever $$ and $f\in S$.
Title  equivalent conditions for uniform integrability 

Canonical name  EquivalentConditionsForUniformIntegrability 
Date of creation  20130322 18:40:17 
Last modified on  20130322 18:40:17 
Owner  gel (22282) 
Last modified by  gel (22282) 
Numerical id  7 
Author  gel (22282) 
Entry type  Theorem 
Classification  msc 28A20 