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Typo fixes in Scheffe.

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Bokuan Li
2026-06-21 00:24:59 -04:00
parent 8c5400bf88
commit 19e2d2df51
1 changed files with 3 additions and 3 deletions
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6
src/fa/lp/ui.tex
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@@ -201,7 +201,7 @@
(UI): Let $\eps > 0$. By (N) and (T), there exists $F_1 \in \fF$ and $A \in \cm$ with $\mu(A) < \infty$ such that for every $f \in F_1$, (UI): Let $\eps > 0$. By (N) and (T), there exists $F_1 \in \fF$ and $A \in \cm$ with $\mu(A) < \infty$ such that for every $f \in F_1$,
\begin{enumerate}[label=(\roman*)] \begin{enumerate}[label=(\roman*)]
\item $|\norm{f}_{L^p(X; E)}^p - \norm{g}_{L^p(X; E)}^p| < \eps$. \item $|\norm{f}_{L^p(X; E)}^p - \norm{g}_{L^p(X; E)}^p| < \eps$.
\item $\int_{A^c}\norm{f}_E^p d\mu < \eps$ and $\int_{A^c}\norm{g}_E^p d\mu$. \item $\int_{A^c}\norm{f}_E^p d\mu, \int_{A^c}\norm{g}_E^p d\mu < \eps$.
\end{enumerate} \end{enumerate}
By (i) and (ii), By (i) and (ii),
@@ -221,7 +221,7 @@
\item $\mu(A \cap \bracs{\norm{f - g}_E > \delta}) < \delta$. \item $\mu(A \cap \bracs{\norm{f - g}_E > \delta}) < \delta$.
\end{enumerate} \end{enumerate}
Let $B \in \cm$ with $B \subset A$ and $\mu(B) < \delta$, then for any $f \in F_3$, Let $B \in \cm$ with $B \subset A$ and $\mu(B) < \delta$, then for any $f \in F_2$,
\begin{align*} \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 \\ \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) \cap \bracs{\norm{f - g}_E \le \delta}}(\norm{g}_E - \delta \vee 0)^pd\mu \\
@@ -250,7 +250,7 @@
so $\int_{B}\norm{f}_E^p d\mu \le 6\eps$ for all $B \in \cm$ with $B \subset A$ and $\mu(B) < \delta$. so $\int_{B}\norm{f}_E^p d\mu \le 6\eps$ for all $B \in \cm$ with $B \subset A$ and $\mu(B) < \delta$.
Finally, by (i) and \hyperref[Markov's Inequality]{theorem:markov-inequality}, there exists $M \ge 0$ such that $\mu\bracs{\norm{f}_E \ge M} < \delta$ for all $f \in F_3$. Therefore for any $f \in F_3$, Finally, by (i) and \hyperref[Markov's Inequality]{theorem:markov-inequality}, there exists $M \ge 0$ such that $\mu\bracs{\norm{f}_E \ge M} < \delta$ for all $f \in F_2$. Therefore for any $f \in F_2$,
\[ \[
\int_{\bracs{\norm{f}_E \ge M}}\norm{f}_E^p d\mu \le \int_{A \cap \bracs{\norm{f}_E \ge M}}\norm{f}_E^p d\mu + \int_{A^c}\norm{f}_E^p d\mu \le 7\eps \int_{\bracs{\norm{f}_E \ge M}}\norm{f}_E^p d\mu \le \int_{A \cap \bracs{\norm{f}_E \ge M}}\norm{f}_E^p d\mu + \int_{A^c}\norm{f}_E^p d\mu \le 7\eps
\] \]