Definition:Uniformly Integrable

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Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $\FF \subseteq \map \MM \Sigma$ be a collection of measurable functions.

Then $\FF$ is said to be uniformly integrable (with respect to $\mu$) if and only if:

$\ds \forall \epsilon > 0: \exists g_\epsilon \in \map {\LL^1_+} \mu: \sup_{f \mathop \in \FF} \int_{\set {\size f \mathop > g_\epsilon} } \size f \rd \mu < \epsilon$


$\map {\LL^1_+} \mu$ is the space of positive $\mu$-integrable functions
$\set {\size f > g_\epsilon}$ is short for $\set {x \in X: \size {\map f x} > \map {g_\epsilon} x}$

Also known as

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