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\begin{proposition}
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\label{proposition:ui-gymnastics}
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Let $(X, \cm, \mu)$ be a measure space, $p \in [1, \infty)$, $E$ be a normed vector space over $K \in \RC$, $\cf \subset L^p(X; K)$, and let $|\cf|^p = \bracsn{\norm{f}^p| f \in \cf}$, then:
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Let $(X, \cm, \mu)$ be a measure space, $p \in [1, \infty)$, $E$ be a normed vector space over $K \in \RC$, $\cf \subset L^p(X; K)$, and let $|\cf|^p = \bracsn{\norm{f}_E^p| f \in \cf}$, then:
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\begin{enumerate}
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\item If $\cf$ is uniformly $p$-integrable, then $|\cf|^p$ is uniformly absolutely continuous with respect to $\mu$.
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\item If $\sup_{f \in \cf}\norm{f}_{L^p(X; E)} < \infty$ and $|\cf|^p$ is uniformly absolutely continuous with respect to $\mu$, then $\cf$ is uniformly $p$-integrable.
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