Definition 15.4.1 (Uniform Integrability).label Let $(X, \cm, \mu)$ be a measure space, $p \in [1, \infty)$, $E$ be a normed vector space over $K \in \RC$, and $\cf \subset L^{p}(X; E)$, then $\cf$ is uniformly $p$-integrable if
\[\lim_{M \to \infty}\sup_{f \in \cf}\int_{\bracs{\norm{f}_E \ge M}}\norm{f}^{p} d\mu = 0\]
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