Let $\mu$ be a positive measure on a measurable space. A collection of functions $\{ f_\alpha \} \subset \Le^1(\mu)$ is uniformly integrable, if for every $\epsilon > 0$ , there exists $\delta > 0$ such that

The usefulness of this definition comes from the Vitali convergence theorem, which uses it to characterize the convergence of functions in $\Le^1(\mu)$ .

Definition in probability theory

In probability theory, a different, and slightly stronger, definition of ``uniform integrability'', is more commonly used:

A collection of functions $\{ f_\alpha \} \subset \Le^1(\mu)$ is uniformly integrable, if for every $\epsilon > 0$ , there exists $t \geq 0$ such that

Properties

1. If a finite number of collections are uniformly integrable, then so is their finite union.
2. A single $f \in \Le^1(\mu)$ is always uniformly integrable.

   To see this, observe that $f$ must be almost everywhere non-infinite. Thus $f \cdot 1_{[ \abs{f} > k]}$ goes to zero a.e. as $k \to \infty$ , and it is bounded by $\abs{f}$ . Then $\int_{[\abs{f} > k]} \abs{f}d\mu \to 0$ by the dominated convergence theorem. Choosing $k$ big enough so that $\int_{[\abs{f} > k]} \abs{f}d\mu < \epsilon$ , and letting $\delta = \epsilon/k$ , we have, when $\mu(E) < \delta$ ,

   
   

Examples

1. If $g$ is an integrable function, then the collection consisting of all measurable functions $f$ dominated by $g$ -- that is, $\abs{f} \leq g$ -- is uniformly integrable.
2. If $X$ is a $\Le^1$ random variable on a probability space $\Omega$ , then the set of all of its conditional expectations,$$ \{ \E[X \mid \mathcal{G}] \colon \mathcal{G}\text{ is a $\sigma$-algebra of $\Omega$} \}\,,$$ is always uniformly integrable.
3. If there is an unbounded increasing function $\phi\colon [0, \infty) \to [0, \infty)$ such that$$ \int \abs{f_\alpha} \phi(\abs{f_\alpha}) \, d\mu$$ is uniformly bounded for all $\alpha$ , then the collection $\{ f_\alpha \}$ is uniformly integrable.