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}[Hahn Decomposition, {{\cite[Theorem 3.3]{Folland}}}]
\begin{theorem}[Hahn Decomposition]
\label{theorem:hahn-decomposition}
Let $(X, \cm)$ be a measurable space and $\mu: \cm \to [-\infty, \infty]$ be a signed measure, then there exists $P, N \in \cm$ such that:
\begin{enumerate}
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The disjoint union $X = P \sqcup N$ is the \textbf{Hahn decomposition} of $\mu$.
\end{theorem}
\begin{proof}
\begin{proof}[Proof {{\cite[Theorem 3.3]{Folland}}}. ]
By flipping the sign of $\mu$, assume without loss of generality that $\mu < \infty$.
(1): Let $M = \sup\bracs{\mu(P)|P \in \cm \text{ positive}}$, then there exists positive sets $\seq{P_n} \subset \cm$ such that $\mu(P_n) \upto M$. Let $P = \bigcup_{n \in \natp}P_n$, then $P$ is positive by \autoref{lemma:positive-sets-properties}, and $\sup_{n \in \natp}\mu(P_n) \le \mu(P) \le M$.