From 346b0bdfd23902ce0bb6597153c7c6f073f799cc Mon Sep 17 00:00:00 2001 From: Bokuan Li Date: Wed, 17 Jun 2026 23:49:24 -0400 Subject: [PATCH] Made more room. --- src/measure/radon/radon.tex | 8 +++++--- 1 file changed, 5 insertions(+), 3 deletions(-) diff --git a/src/measure/radon/radon.tex b/src/measure/radon/radon.tex index f27303e..84d2182 100644 --- a/src/measure/radon/radon.tex +++ b/src/measure/radon/radon.tex @@ -178,9 +178,11 @@ By \autoref{proposition:lp-simple-dense}, $\Sigma(X, \cm; E) \cap L^p(X; E)$ is dense in $L^p(X; E)$. Using linearity, it is sufficient to approximate indicator functions of Borel sets with finite measure. Let $A \in \cb_X$ and $\eps > 0$. By \autoref{proposition:radon-regular-sigma-finite}, there exists $U \in \cn^o(A)$ and $K \subset A$ compact such that $\mu(U \setminus A), \mu(A \setminus K) < \eps/2$. By \hyperref[Urysohn's lemma]{lemma:lch-urysohn}, there exists $f \in C_c(X; [0, 1])$ such that $f|_K = 1$ and $\supp{f} \subset U$. In which case, for any $y \in E$, - \[ - \norm{x \cdot \one_A - x \cdot f}_{L^p(X; E)} \le \norm{x}_E \mu(\bracs{f \ne \one_A})^{1/p} \le \norm{x}_E\mu(U \setminus K)^{1/p} < \eps^{1/p}\norm{x}_E - \] + \begin{align*} + \norm{x \cdot \one_A - x \cdot f}_{L^p(X; E)} &\le \norm{x}_E \mu(\bracs{f \ne \one_A})^{1/p} \\ + &\le \norm{x}_E\mu(U \setminus K)^{1/p} < \eps^{1/p}\norm{x}_E + \end{align*} + \end{proof} \begin{theorem}[Lusin]