Added basic theory of Lp spaces, alongside some integral stuff.

src/measure/bochner-integral/index.tex

@@ -0,0 +1,4 @@
+\chapter{Bochner Integral}
+\label{chap:bochner-integral}
+
+\input{./src/measure/bochner-integral/strongly.tex}

src/measure/bochner-integral/strongly.tex

@@ -0,0 +1,27 @@
+\section{Strongly Measurable Functions}
+\label{section:strongly-measurable}
+
+\begin{definition}[Strongly Measurable Function]
+\label{definition:strongly-measurable}
+    Let $(X, \cm)$ be a measurable space, $E$ be a normed vector space over $K \in \RC$, and $f: X \to E$, then the following are equivalent:
+    \begin{enumerate}
+        \item For each $\phi \in E^*$, $\phi \circ f$ is $(\cm, \cb_K)$-measurable and $f(X) \subset E$ is separable.
+        \item $f$ is $(\cm, \cb_E)$ measurable and $f(X) \subset E$ is separable.
+        \item There exists a sequence $\seq{f_n} \subset \Sigma(X, \cm; E)$ such that
+        \begin{enumerate}
+            \item[(a)] For each $n \in \natp$, $\norm{f_n}_E \le \norm{f}_E$.
+            \item[(b)] $\norm{f_n(x) - f(x)}_E \to 0$ pointwise as $n \to \infty$.
+        \end{enumerate}
+    \end{enumerate}
+\end{definition}
+\begin{proof}
+    (1) $\Rightarrow$ (2): TODO
+
+    (2) $\Rightarrow$ (3): By \ref{proposition:measurable-simple-separable-norm}.
+
+    (3) $\Rightarrow$ (1): For each $\phi \in E^*$, $\phi \circ f = \limv{n}\phi \circ f_n$ is measurable by \ref{proposition:limit-measurable}. Since
+    \[
+    f(X) \subset \ol{\bigcup_{n \in \natp}f_n(X)}
+    \]
+    and each $f_n$ is finitely-valued, $f(X)$ is separable.
+\end{proof}

src/measure/index.tex

@@ -5,3 +5,4 @@
 \input{./src/measure/measure/index.tex}
 \input{./src/measure/measurable-maps/index.tex}
 \input{./src/measure/lebesgue-integral/index.tex}
+\input{./src/measure/bochner-integral/index.tex}

src/measure/measure/radon.tex

@@ -131,7 +131,17 @@
     and $\mu = \nu$.
 
     Since $X$ is $\sigma$-compact, $\mu$ is $\sigma$-finite, so $\mu$ is regular by \ref{proposition:radon-regular-sigma-finite}.
+\end{proof}
+
+\begin{proposition}
+\label{proposition:radon-cc-dense}
+    Let $X$ be a LCH space, $\mu: \cb_X \to [0, \infty]$ be a Radon measure, $E$ be a normed space, and $p \in [1, \infty)$, then $C_c(X; E)$ is dense in $L^p(X; E)$.
+\end{proposition}
+\begin{proof}
+    By \ref{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 \ref{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 Urysohn's lemma (\ref{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 $y \in E$,
     \[
-    \color{blue}{\int}\llap{\color{green}{\int}}
+    \norm{x \cdot \one_A - x \cdot f}_{L^p(X; E)} \le \norm{x}_E \mu(\bracs{f \ne \one_A})^{1/p} \le \norm{x}_E\mu(U \setminus K)^{1/p} < \eps^{1/p}\norm{x}_E
     \]
 \end{proof}