diff --git a/src/fa/lp/ui.tex b/src/fa/lp/ui.tex index f0d1e37..f6143a6 100644 --- a/src/fa/lp/ui.tex +++ b/src/fa/lp/ui.tex @@ -13,7 +13,7 @@ \begin{proposition} \label{proposition:ui-gymnastics} - 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: + 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: \begin{enumerate} \item If $\cf$ is uniformly $p$-integrable, then $|\cf|^p$ is uniformly absolutely continuous with respect to $\mu$. \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.