Polished A-A and added new lines for broken enumerates.
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Bokuan Li
2026-05-05 01:50:35 -04:00
parent 47a7e1de68
commit 0f2e69d1f9
81 changed files with 441 additions and 185 deletions

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@@ -52,7 +52,8 @@
\begin{enumerate}
\item[(a)] $f_n \to f$ strongly pointwise.
\item[(b)] There exists $g \in L^1(X) \cap L^+(X)$ such that $\norm{f_n}_E \le g$ for all $n \in \natp$.
\end{enumerate}
\end\{enumerate\}
then $\int f d\mu = \limv{n}\int f_n d\mu$.
\end{theorem}
@@ -76,7 +77,8 @@
\begin{enumerate}
\item For any $x \in E$ and $A \in \cm$, $I_\lambda(x \cdot \one_A) = x \mu(A)$.
\item For any $f \in L^1(X, |\mu|; E)$, $\normn{I_\lambda f}_{G} \le \norm{\lambda}_{L^2(E, F; G)} \cdot \norm{f}_{L^1(X, |\mu|; E)}$.
\end{enumerate}
\end\{enumerate\}
For any $f \in L^1(X; E)$, $I_\lambda f = \int \lambda(f, d\mu)$ is the \textbf{Bochner integral} of $f$ with respect to $\mu$ and $\lambda$.
\end{definition}
@@ -96,7 +98,8 @@
\begin{enumerate}
\item[(a)] $f_n \to f$ strongly pointwise.
\item[(b)] There exists $g \in L^1(X) \cap L^+(X)$ such that $\norm{f_n}_E \le g$ for all $n \in \natp$.
\end{enumerate}
\end\{enumerate\}
then $\int \lambda(f , d\mu) = \limv{n}\int \lambda(f_n, d\mu)$.
\end{theorem}