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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@@ -76,7 +76,8 @@
\item[(a)] For each $y \in Y$, $y \in \ol{N(y)^o}$.
\item[(b)] $\bigcap_{y \in Y}N(y) \ne \emptyset$.
\item[(c)] For any $y_0 \in Y$, $\bracs{y \in Y|y_0 \in N(y)} \in \cb_Y$.
\end{enumerate}
\end\{enumerate\}
Then, for any $f: X \to Y$, the following are equivalent:
\begin{enumerate}
\item $f$ is $(\cm, \cb_Y)$-measurable.
@@ -84,7 +85,8 @@
\begin{enumerate}
\item[(i)] For each $x \in X$ and $n \in \natp$, $f_n(x) \in N(f(x))$.
\item[(ii)] $f_n \to f$ pointwise.
\end{enumerate}
\end\{enumerate\}
\item There exists a sequence $\seq{f_n}$ of $(\cm, \cb_Y)$-measurable simple functions such that $f_n \to f$ pointwise.
\end{enumerate}
\end{proposition}
@@ -146,6 +148,7 @@
\bracs{y \in E|y_0 \in N(y)} = \bracs{y \in E|\norm{y_0}_E \le \norm{y}_E} \in \cb_E
\]
\end{enumerate}
\end\{enumerate\}
By \autoref{proposition:measurable-simple-separable}, (1) and (2) are equivalent.
\end{proof}