---
tag: FR210Q35CB
kind: definition
name: Uniform integrability
msc: [28A25, 60A10]
defines: VRVVKMXDGV
uses: [63Z1Q4FS23, X7GTF3D98F, DG76G5Z6XW]
sources:
  - url: https://en.wikipedia.org/wiki/Uniform_integrability
---

## Definition

Let $(X, \Sigma, \mu)$ be a [[finite measure|63Z1Q4FS23]] space.
Let $\mathcal{F}$ be a family of real or complex [[integrable functions|X7GTF3D98F]] on $X$.
The family $\mathcal{F}$ is *uniformly integrable* if
$$\lim_{M \to \infty} \ \sup_{f \in \mathcal{F}} \int_{\set{\abs{f} > M}} \abs{f} \, d\mu = 0,$$
where the integrals are [[integrals of non-negative functions|DG76G5Z6XW]].

## Remarks

Example: on $[0, 1]$ with Lebesgue measure, let $\mathcal{F} = \set{f : \abs{f} \le 2}$.
For $M \ge 2$ each set $\set{\abs{f} > M}$ is empty, so the supremum is $0$.

Non-example: on $[0, 1]$, let $f_n = n \mathbf{1}_{(0, 1/n)}$.
For $n > M$ the integral of $f_n$ over $\set{f_n > M}$ equals $1$, so the supremum is $1$ for every $M$.

On a finite measure space the condition is equivalent to two conditions together.
First, $\sup_{f \in \mathcal{F}} \int \abs{f} \, d\mu < \infty$.
Second, for every $\varepsilon > 0$ there is $\delta > 0$ with $\int_A \abs{f} \, d\mu < \varepsilon$ for all $f \in \mathcal{F}$ whenever $\mu(A) < \delta$.
Any family dominated by one integrable function is uniformly integrable.
Bounded families in $L^p$ for some $p > 1$ are uniformly integrable.
On infinite measure spaces other versions of the definition are in use.