From 77770954611509f4bf464d7f75b3a08157dfeb34 Mon Sep 17 00:00:00 2001 From: Bokuan Li Date: Wed, 24 Jun 2026 23:28:30 -0400 Subject: [PATCH] Added citations for elements taken from Cohn's Measure Theory. --- refs.bib | 12 ++++++++++++ src/measure/measurable-maps/metric.tex | 4 ++-- 2 files changed, 14 insertions(+), 2 deletions(-) diff --git a/refs.bib b/refs.bib index 88a6bfb..b45cf36 100644 --- a/refs.bib +++ b/refs.bib @@ -235,4 +235,16 @@ year = {2013}, doi = {10.1007/978-1-4471-4820-3}, isbn = {978-1-4471-4820-3} +} + +@book{CohnMeasure, + author = {Cohn, Donald L.}, + title = {Measure Theory}, + edition = {2nd}, + series = {Birkhäuser Advanced Texts Basler Lehrbücher}, + publisher = {Birkhäuser}, + address = {New York}, + year = {2013}, + isbn = {978-1-4614-6955-1}, + doi = {10.1007/978-1-4614-6956-8} } \ No newline at end of file diff --git a/src/measure/measurable-maps/metric.tex b/src/measure/measurable-maps/metric.tex index 83d4b1f..5cb0539 100644 --- a/src/measure/measurable-maps/metric.tex +++ b/src/measure/measurable-maps/metric.tex @@ -40,7 +40,7 @@ \item For each $\phi \in C(X; [0, 1])$, $\phi \circ f$ is $(\cm, \cb_\real)$-measurable. \end{enumerate} \end{proposition} -\begin{proof} +\begin{proof}[Proof, {{\cite[Lemma 8.1.9]{CohnMeasure}}}. ] (2) $\Rightarrow$ (1): For each $U \subset X$ open, the function \[ d_{U^c}: X \to [0, 1] \quad x \mapsto d(x, U^c) \wedge 1 @@ -57,7 +57,7 @@ \item If $f = \limv{n}f_n$ exists, then it is $(\cm, \cb_Y)$-measurable. \end{enumerate} \end{proposition} -\begin{proof} +\begin{proof}[Proof, {{\cite[Proposition 8.1.10-8.1.11]{CohnMeasure}}}. ] (1): Let $d$ be a complete metric on $Y$, then for any $x \in X$, $\limv{n}f_n(x)$ exists if and only if $\seq{f_n(x)}$ is Cauchy. In which case, \[ \bracs{\limv{n}f_n \text{ exists}} = \bigcap_{k \in \natp}\bigcup_{N \in \natp}\bigcap_{m \in \natp}\bigcap_{n \in \natp}\bracsn{d(f_m, f_n) < 1/k}