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Added a variation bound for scalar measures.

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Bokuan Li
2026-06-19 12:59:33 -04:00
parent 9504125410
commit 64788a5322
1 changed files with 21 additions and 0 deletions
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src/measure/vector/variation.tex
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@@ -90,3 +90,24 @@
Let $(X, \cm)$ be a measurable space and $\mu$ be a signed/vector measure, then $\mu$ is \textbf{finite} if $|\mu|$ is finite. Let $(X, \cm)$ be a measurable space and $\mu$ be a signed/vector measure, then $\mu$ is \textbf{finite} if $|\mu|$ is finite.
\end{definition} \end{definition}
\begin{lemma}
\label{lemma:variation-bound}
Let $(X, \cm)$ be a measurable space, $K \in \RC$, $\mu: \cm \to K$ be a signed/complex measure, then:
\begin{enumerate}
\item If $K = \real$, then $|\mu|(X) \le 2\sup_{A \in \cm}\norm{\mu(A)}_E$.
\item If $K = \complex$, then $|\mu|(X) \le 4\sup_{A \in \cm}\norm{\mu(A)}_E$.
\end{enumerate}
\end{lemma}
\begin{proof}
(1, scalar): Let $X = P \sqcup N$ with $P, N \in \cm$ be a \hyperref[Hahn decomposition]{theorem:hahn-decomposition} of $\mu$, then
\[
|\mu|(X) = |\mu(X \cap P)| + |\mu(X \cap N)| \le 2\sup_{A \in \cm}\norm{\mu(A)}_E
\]
(2, scalar): By (1),
\[
|\mu|(X) \le |\text{Re}(\mu)|(X) + |\text{Im}(\mu)|(X) \le 4\sup_{A \in \cm}\norm{\mu(A)}_E
\]
\end{proof}