Fixed more typos for Scheffe.

src/fa/lp/ui.tex

@@ -159,9 +159,9 @@
 
 \begin{lemma}[Scheffé]
 \label{lemma:scheffe}
-    Let $(X, \cm, \mu)$ be a measure space, $p \in [1, \infty)$, $E$ be a normed vector space over $K \in \RC$, $\fF \subset 2^{L^p(X; E)}$ be a filter, and $g \in L^p(X; E)$, then $\fF \to f$ in $L^p(X; E)$ if and only if:
+    Let $(X, \cm, \mu)$ be a measure space, $p \in [1, \infty)$, $E$ be a normed vector space over $K \in \RC$, $\fF \subset 2^{L^p(X; E)}$ be a filter, and $g \in L^p(X; E)$, then $\fF \to g$ in $L^p(X; E)$ if and only if:
     \begin{enumerate}
-        \item[(M)] $\fF \to g$ is locally in measure. 
+        \item[(M)] $\fF \to g$ locally in measure. 
         \item[(N)] $\lim_{f, \fF} \norm{f}_{L^p(X; E)} = \norm{g}_{L^p(X; E)}$. 
     \end{enumerate}
 \end{lemma}
@@ -206,7 +206,7 @@
     
     By (i) and (ii), 
     \begin{enumerate}[label=(\roman*), start=2]
-        \item $\abs{\int_{A}\norm{f}_E^p d\mu - \int_{A}\norm{g}_E^p} \le 3\eps$. 
+        \item $\abs{\int_{A}\norm{f}_E^p d\mu - \int_{A}\norm{g}_E^p}d\mu \le 3\eps$. 
     \end{enumerate}
     
 
@@ -225,10 +225,10 @@
     \begin{align*}
         \int_{A \setminus B}\norm{f}_E^p d\mu &\ge \int_{(A \setminus B) \cap \bracs{\norm{f - g}_E \le \delta}}\norm{f}_E^p d\mu \\
         &\ge \int_{(A \setminus B) \cap \bracs{\norm{f - g}_E \le \delta}}(\norm{g}_E - \delta \vee 0)^pd\mu \\
-        &\ge \int_{(A \setminus B)}(\norm{g}_E - \delta \vee 0)^pd\mu - \int_{\bracs{\norm{f - g}_E > \delta}} \norm{g}_E^p d\mu
+        &\ge \int_{(A \setminus B)}(\norm{g}_E - \delta \vee 0)^pd\mu - \int_{A \cap \bracs{\norm{f - g}_E > \delta}} \norm{g}_E^p d\mu
     \end{align*}
 
-    By (vi) and (iv), $\int_{\bracs{\norm{f - g}_E > \delta}} \norm{g}_E^p d\mu < \eps$, so
+    By (vi) and (iv), $\int_{A \cap \bracs{\norm{f - g}_E > \delta}} \norm{g}_E^p d\mu < \eps$, so
     \[
     \int_{A \setminus B}\norm{f}_E^p d\mu \ge \int_{(A \setminus B)}(\norm{g}_E - \delta \vee 0)^pd\mu - \eps
     \]