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Fixed awkward phrasing.
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Bokuan Li
2026-06-16 13:43:42 -04:00
parent 76107709a8
commit c31994b247
1 changed files with 1 additions and 1 deletions
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src/measure/vector/fin.tex
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@@ -103,7 +103,7 @@ Despite the fact that it does not cover the full dual space, the bounded Borel f
\item[(P)] For each $x \in X$, $\bracs{x} \in \cm$, and the delta mass $\delta_x$ is in $\mathscr{M}$. \item[(P)] For each $x \in X$, $\bracs{x} \in \cm$, and the delta mass $\delta_x$ is in $\mathscr{M}$.
\end{enumerate} \end{enumerate}
then for any sequence $f_n: X \to E^*$ of bounded measurable functions and $f: X \to E^*$ be a bounded measurable function, the following are equivalent: Then, for any bounded measurable functions $\bracsn{f_n: X \to E^*|n \in \natp}$ and $f: X \to E^*$, the following are equivalent:
\begin{enumerate} \begin{enumerate}
\item For each $\mu \in \mathscr{M}$, $\limv{n}\int f_n d\mu = \int f d\mu$. \item For each $\mu \in \mathscr{M}$, $\limv{n}\int f_n d\mu = \int f d\mu$.
\item For each $x \in X$, $\limv{n}f_n(x) = f(x)$, and $\sup_{n \in \natp}\norm{f_n}_u < \infty$. \item For each $x \in X$, $\limv{n}f_n(x) = f(x)$, and $\sup_{n \in \natp}\norm{f_n}_u < \infty$.