Definition:Uniformly Integrable

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Definition

Let $\left({X, \Sigma, \mu}\right)$ be a measure space.

Let $\mathcal F \subseteq \mathcal M \left({\Sigma}\right)$ be a collection of measurable functions.


Then $\mathcal F$ is said to be uniformly integrable (with respect to $\mu$) iff:

$\forall \epsilon > 0: \exists g_\epsilon \in \mathcal{L}^1_+ \left({\mu}\right): \displaystyle \sup_{f \mathop \in \mathcal F} \int_{\left\{{\left\vert{f}\right\vert > g_\epsilon}\right\}} \left\vert{f}\right\vert \, \mathrm d \mu < \epsilon$

where:

$\mathcal{L}^1_+ \left({\mu}\right)$ is the space of positive $\mu$-integrable functions
$\left\{{\left\vert{f}\right\vert > g_\epsilon}\right\}$ is short for $\left\{{x \in X: \left\vert{f \left({x}\right)}\right\vert > g_\epsilon \left({x}\right)}\right\}$


Also known as

Some authors refer to uniformly integrable collections $\mathcal F$ as being equi-integrable.


Sources