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Added elements of localisable measures.

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Bokuan Li
2026-06-29 20:55:01 -04:00
parent 831acc66cc
commit 38099f1b19
4 changed files with 124 additions and 2 deletions
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src/measure/vector/rn.tex
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@@ -1,7 +1,7 @@
\section{The Lebesgue-Radon-Nikodym Theorem}
\label{section:lebesgue-radon-nikodym}
\begin{theorem}[Lebesgue-Radon-Nikodym]
\begin{theorem}[Lebesgue-Radon-Nikodym (Finite)]
\label{theorem:lebesgue-radon-nikodym}
Let $(X, \cm)$ be a measurable space, $\mu$ be a $\sigma$-finite positive measure on $(X, \cm)$, $H$ be a Hilbert space over $K \in \RC$, and $\nu: \cm \to H$ be a finite vector measure, then there exists a unique pair of finite vector measures $\nu_a, \nu_s: \cm \to H$ such that:
\begin{enumerate}
@@ -96,3 +96,4 @@
(Uniqueness): For any decomposition $\nu = \rho_a + \rho_s$ satisfying the above, then $\rho_a - \nu_a = \nu_s - \rho_s$ with $\rho_a - \nu_a \perp \nu_s - \rho_s$. Therefore $\rho_a = \nu_a$, $\rho_s = \nu_s$, and the decomposition is unique.
\end{proof}