Adjusted citation formats. Moved citation off of named theorems if possible.
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Bokuan Li
2026-03-19 23:58:16 -04:00
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\end{proof}
\begin{theorem}[Riesz Representation Theorem, {{\cite[Theorem 7.2]{Folland}}}]
\begin{theorem}[Riesz Representation Theorem]
\label{theorem:riesz-radon}
Let $(X, \topo)$ be a LCH space and $I \in \hom(C_c(X; \real); \real)$ be a positive linear functional, then there exists a Borel measure $\mu: \cb_X \to [0, \infty]$ such that:
\begin{enumerate}
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\item[(U)] If $\nu: \cb_X \to [0, \infty]$ is a Borel measure that satisfies (3) and (4), then $\mu = \nu$.
\end{enumerate}
\end{theorem}
\begin{proof}
\begin{proof}[Proof {{\cite[Theorem 7.2]{Folland}}}. ]
(1): For any $U \in \topo$ and $f \in C_c(X; \real)$, denote $f \prec U$ if $f \in C_c(X; [0, 1])$ and $\supp{f} \subset U$. Let
\[
\mu_0: \topo \to [0, \infty] \quad U \mapsto \sup_{f \prec U}\dpb{f, I}{C_c(X; \real)}