Added existence of the Haar measure.

src/measure/lcg/haar.tex

@@ -0,0 +1,152 @@
+\section{Haar Measures}
+\label{section:haar}
+
+\begin{definition}[Haar Measure]
+\label{definition:haar-measure}
+    Let $G$ be a locally compact group and $\mu: \cb_G \to [0, \infty]$ be a non-zero Radon measure, then $\mu$ is a \textbf{left Haar measure} if
+    \begin{enumerate}
+        \item[(LH)] For each $g \in G$ and $A \in \cb_G$, $\mu(gA) = \mu(A)$. 
+    \end{enumerate}
+
+    and a \textbf{right Haar measure} if
+    \begin{enumerate}
+        \item[(RH)] For each $g \in G$ and $A \in \cb_G$, $\mu(Ag) = \mu(A)$. 
+    \end{enumerate}
+\end{definition}
+
+\begin{lemma}
+\label{lemma:lc-sigma-compact}
+    Let $G$ be a locally compact group, then there exists an open and closed subgroup $H$ that is $\sigma$-compact. 
+\end{lemma}
+\begin{proof}
+    Let $K \in \cn_G(1)$ and $K^{(1)} = K$. For each $n \in \natp$, let $K^{(n+1)} = KK^{(n)}$, then $K^{(n+1)}$ is compact by \autoref{proposition:compact-extensions} with $K^{(n+1)} \in \cn_G(K^{(n)})$. Let $H = \bigcup_{n \in \natp}K^{(n)}$, then $H$ is a subgroup of $G$, which is open by \autoref{lemma:openneighbourhood}. Since $H$ admits an exhaustion by compact sets, it is $\sigma$-compact. 
+
+    Finally, since
+    \[
+    G \setminus H = G \setminus \bigcup_{n \in \natp} K^{(n)} = \bigcap_{n \in \natp}G \setminus K^{(n)}
+    \]
+
+    and $K^{(n)}$ is closed for each $n \in \natp$ by \autoref{proposition:compact-closed}, $G \setminus H$ is closed, and hence $H$ is open. 
+\end{proof}
+
+
+\begin{definition}[Covering Ratio]
+\label{definition:lcg-covering-ratio}
+    Let $G$ be a locally compact group and $f, g \in C_c^+(G)$, then
+    \[
+    (f: g) = \inf\bracs{\sum_{j = 1}^n c_j \bigg | \seqf{c_j} \subset [0, \infty), \seqf{x_j} \subset G, f \le \sum_{j = 1}^n c_j L_{x_j}g}
+    \]
+
+    is the \textbf{covering ratio} of $f$ by $g$. 
+\end{definition}
+
+\begin{proposition}
+\label{proposition:covering-ratio-gymnastics}
+    Let $G$ be a locally compact group and $f, h, g \in C_c^+(G)$, then:
+    \begin{enumerate}
+        \item If $g \ne 0$, then $(f: g) < \infty$. 
+        \item $(f, h: g) \le (h: g) + (h: g)$. 
+        \item For each $\lambda \ge 0$, $(\lambda f: g) = \lambda(f: g)$. 
+        \item If $f \le h$, then $(f: g) \le (h: g)$. 
+        \item $(f: g) \le (f: h)(h: g)$. 
+        \item $(f: g) \ge \norm{f}_u/\norm{g}_h$. 
+        \item For each $x \in G$, $(L_xf: g) = (f: g)$. 
+    \end{enumerate}
+\end{proposition}
+% Proof omitted due to obviousness. 
+
+\begin{lemma}
+\label{lemma:haar-approx}
+    Let $G$ be a locally compact group, $f, f'\in C_c^+(G)$, and $\eps > 0$, then there exists $V \in \cn_G(1)$ such that for any $g \in C_c^+(V)$ with $g \ne 0$, 
+    \[
+    (f: g) + (f': g) \le (f + f': g) + \eps
+    \]
+\end{lemma}
+\begin{proof}[Proof, {{\cite[Lemma 2.18]{FollandHarmonic}}}. ]
+    By \hyperref[Urysohn's Lemma]{lemma:lch-urysohn}, there exists $\eta \in C_c^+(G; [0, 1])$ such that $\eta|_{\supp{f} \cup \supp{f'}} = 1$. 
+
+    Let $\delta > 0$, and define
+    \[
+    H = f + f' + \delta \eta \quad h = \frac{f}{H} \quad h' = \frac{f'}{H}
+    \]
+    
+    By \autoref{proposition:lcg-cc-uc}, there exists $V \in \cn_G(1)$ such that for any $x, y \in G$ with $x^{-1}y \in V$, 
+    \[
+    |h(x) - h(y)|, |h'(x) - h'(y)| < \delta
+    \] 
+
+    Let $g \in C_c^+(V)$, $\seqf{c_j} \subset [0, \infty)$, and $\seqf{x_j} \subset G$ such that $H \le \sum_{j = 1}^n c_j L_{x_j}\phi$, then for each $x \in G$, 
+    \begin{align*}
+        f(x) &= H(x)h(x) \le \sum_{j = 1}^n c_j L_{x_j}g(x)h(x) = \sum_{j = 1}^n c_jg(x_j^{-1}x)h(x) \\
+        &\le \sum_{j = 1}^n c_j[h(x_j) + \delta] \cdot  L_{x_j}g(x)
+    \end{align*}
+
+    Likewise, 
+    \[
+    f'(x) \le \sum_{j = 1}^n c_j[h'(x_j) + \delta] \cdot L_{x_j}g(x)
+    \]
+
+    As $h + h' \le 1$, 
+    \[
+        (f: g) + (f': g) \le \sum_{j = 1}^n c_j[h(x_j) + h'(x_j) + 2\delta]
+    \]
+    
+    Since the above holds for all such $\seqf{c_j} \subset [0, \infty)$ and $\seqf{x_j} \subset G$, 
+    \begin{align*}
+    (f: g) + (f': g) &\le (1 + 2\delta)(H: g) \\
+    &\le (1 + 2\delta)[(f + f': g) + \delta(\eta: g)]
+    \end{align*}
+\end{proof}
+
+\begin{theorem}[Haar]
+\label{theorem:haar}
+    Let $G$ be a locally compact group, then:
+    \begin{enumerate}
+        \item There exists a left/right Haar measure on $G$. 
+        \item For any two left/right Haar measures $\mu$ and $\nu$ on $G$, there exists $\lambda > 0$ such that $\mu = \lambda \nu$. 
+    \end{enumerate}
+\end{theorem}
+\begin{proof}[Proof, {{\cite[Theorem 2.10, 2.20]{FollandHarmonic}}}. ]
+    (1): Fix $h \in C_c^+(G)$ with $h \ne 0$. For each $g \in C_c^+(G)$ with $g \ne 0$, let
+    \[
+    I_g: C_c^+(G) \to [0, \infty) \quad f \mapsto \frac{(f: g)}{(h: g)}
+    \]
+
+    then by (5) of \autoref{proposition:covering-ratio-gymnastics}, for each $f \in C_c^+(G)$ with $f \ne 0$, 
+    \[
+    \frac{1}{(h: f)} = \frac{(f: g)}{(h: f)(f: g)} \le I_g(f) \le \frac{(f: h)(h: g)}{(h: g)} = (f: h)
+    \]
+
+    Thus $\mathcal{I}(f) = \bracs{I_g(f)|g \in C_c^+(G) \setminus \bracs{0}}$ is precompact for each $f \in C_c^+(G)$. 
+
+    For each $V \in \cn_G(1)$, let $E_V = \bracs{I_g|g \in C_c^+(V) \setminus \bracs{0}}$, then $\fF = \bracs{E_V|V \in \cn_G(1)}$ is a filter on the product space $\prod_{f \in C_c^+(G)}\ol{\mathcal{I}(f)}$. By \hyperref[Tychonoff's Theorem]{theorem:tychonoff}, there exists $\bigcap_{V \in \cn_G(1)}\ol{E_V} \ne \emptyset$. 
+
+    Let $I \in \bigcap_{V \in \cn_G(1)}\ol{E_V}$, then by continuity, 
+    \begin{enumerate}[label=(\roman*)]
+        \item For each $f \in C_c^+(G) \setminus \bracs{0}$, $I(f) \in [(h: f)^{-1}, (f: h)]$. 
+        \item For every $\lambda \ge 0$ and $f \in C_c^+(G)$, $I(\lambda f) = \lambda I(f)$.
+        \item For any $x \in G$, $I(L_xf) = I(f)$. 
+        \item For each $f, f' \in C_c^+(G)$, $I(f + g) \le I(f) + I(g)$. 
+    \end{enumerate}
+    
+    Let $f, f' \in C_c^+(G)$ and $\eps > 0$. By \autoref{lemma:haar-approx}, there exists $V \in \cn_G(1)$ such that for each $g \in E_V$, 
+    \begin{enumerate}[label=(\alph*)]
+        \item $|I_g(f) - I(f)|, |I_g(f') - I(f')| < \eps$. 
+        \item $I_g(f) + I_g(f') \le I_g(f + f') + \eps$. 
+    \end{enumerate}
+
+    In which case, $I(f) + I(f') \le I(f + f') + \eps$. Since this holds for all $\eps > 0$, 
+    \begin{enumerate}[start=4, label=(\roman*)]
+        \item For each $f, f' \in C_c^+(G)$, $I(f + g) \ge I(f) + I(g)$. 
+    \end{enumerate}
+
+    Using \autoref{lemma:positive-functional-extension}, $I$ extends to a positive linear functional on $C_c(G; \real)$, with $I(f) > 0$ for all $f \in C_c^+(G) \setminus \bracs{0}$, and
+    \begin{enumerate}
+        \item[(LH)] For each $f \in C_c(G)$ and $x \in G$, $I(L_xf) = I(f)$. 
+    \end{enumerate}
+    
+    By (i) and the \hyperref[Riesz Representation Theorem]{theorem:riesz-radon}, there exists a unique non-zero Radon measure $\mu: \cb_G \to [0, \infty]$ such that for each $f \in C_c^+(G)$, $I(f) = \int_G f d\mu$. Finally, by \hyperref[density of $C_c(\mu; \real)$ in $L^1(\mu; \real)$]{proposition:radon-cc-dense} and (LH), $\mu$ is a left Haar measure. 
+\end{proof}
+
+
+

src/measure/lcg/lcg.tex

@@ -6,4 +6,10 @@
     Let $G$ be a topological group, then $G$ is \textbf{locally compact} if $G$ is a LCH space. 
 \end{definition}
 
-
+\begin{proposition}
+\label{proposition:lcg-cc-uc}
+    Let $G$ be a locally compact group, $E$ be a TVS over $K \in \RC$, and $\phi \in C_c(G; E)$, then $\phi$ is left and right uniformly continuous. 
+\end{proposition}
+\begin{proof}
+    By \autoref{lemma:lch-compact-neighbour}, there exists a compact neighbourhood $U \in \cn_G(\text{supp}(\phi))$. By \autoref{proposition:uniform-continuous-compact}, $\phi|_{U}$ and $\phi|_{\supp(\phi)^c}$ are both left and right uniformly continuous. Therefore $\phi$ is left and right uniformly continuous. 
+\end{proof}

src/topology/main/lch.tex

@@ -40,7 +40,7 @@
 \label{lemma:lch-urysohn}
     Let $X$ be a LCH space, $K \subset X$ be compact, and $U \in \cn(K)$, then there exists $f \in C_c(X; [0, 1])$ such that $\supp{f} \subset U$.
 \end{lemma}
-\begin{proof}[Proof {{\cite[Lemma 4.32]{Folland}}}. ]
+\begin{proof}[Proof, {{\cite[Lemma 4.32]{Folland}}}. ]
     By \autoref{lemma:lch-compact-neighbour}, there exists $V, W \in \cn^o(K)$ precompact such that
     \[
     K \subset V \subset \ol{V} \subset W \subset \ol{W} \subset U
@@ -106,7 +106,7 @@
 
 
 
-\begin{proposition}[{{\cite[Proposition 4.39]{Folland}}}]
+\begin{proposition}
 \label{proposition:lch-sigma-compact}
     Let $X$ be a LCH space, then the following are equivalent:
     \begin{enumerate}
@@ -114,7 +114,7 @@
         \item There exists an exhaustion of $X$ by compact sets.
     \end{enumerate}
 \end{proposition}
-\begin{proof}
+\begin{proof}[Proof, {{\cite[Proposition 4.39]{Folland}}}. ]
     (1) $\Rightarrow$ (2): Let $\seq{K_n} \subset 2^X$ be compact such that $\bigcup_{n \in \natp}K_n = X$, and $U_0 = \emptyset$.
 
     Assume inductively that $\bracs{U_j}_0^n$ has been constructed such that:
@@ -132,11 +132,11 @@
     Thus $\bracs{U_j}_0^{n+1}$ satisfies (a), (b), and (c), and $\seq{U_n}$ is an exhaustion of $X$ by compact sets.
 \end{proof}
 
-\begin{proposition}[{{\cite[Proposition 4.41]{Folland}}}]
+\begin{proposition}
 \label{proposition:lch-partition-of-unity}
     Let $X$ be a LCH space, $K \subset X$ be compact, and $\seqi{U}$ be an open cover of $K$, then there exists a $C_c$ partition of unity on $K$ subordinate to $\seqi{U}$.
 \end{proposition}
-\begin{proof}
+\begin{proof}[Proof, {{\cite[Proposition 4.41]{Folland}}}. ]
     Since $K$ is compact, assume without loss of generality that $\seqi{U} = \seqf{U_j}$.
 
     For every $x \in K$, there exists $1 \le j \le n$ and $N_x \in \cn(x)$ compact such that $x \in N_x \subset U_j$. By compactness of $K$, there exists $\seqf[m]{x_j} \subset K$ such that $K = \bigcup_{j = 1}^m N_{x_j}$.
@@ -260,6 +260,14 @@
     (6) $\Rightarrow$ (2): Let $\seqi{f} \subset C_c(X; [0, 1])$ be a partition of unity. For each $i \in I$, let $V_i = \bracs{f_i > 0}$, then $\seqi{V}$ is a locally finite precompact open cover of $\mathcal{U}$.
 \end{proof}
 
+\begin{proposition}
+\label{proposition:paracompact-lch-normal}
+    Let $X$ be a paracompact LCH space, then $X$ is normal. 
+\end{proposition}
+\begin{proof}
+    Let $A, B \subset X$ be disjoint closed sets. By \autoref{proposition:lch-paracompact}, there exists a partition of unity $\seqi{f}$ subordinate to $\bracs{A^c, B^c}$. Let $I = I_A \sqcup I_B$ such that for each $i \in I_A$, $\supp{f_i} \subset B^c$, and for each $i \in I_B$, $\supp{f_i} \subset A^c$. Take $f = \sum_{i \in I_A}f_i$ and $g = \sum_{i \in I_B}f_i$, then since $f|_B = 0$ and $g|_A = 0$, $\bracs{f \ge 2/3} \in \cn_X(A)$ and $\bracs{f \le 1/3} \in \cn_X(B)$. 
+\end{proof}
+
 \begin{proposition}
 \label{proposition:lch-sigma-paracompact}
     Let $X$ be a $\sigma$-compact LCH space, then $X$ is paracompact.
@@ -270,4 +278,3 @@
     Let $x \in X$, then there exists $n \in \natp$ such that $x \in U_n \setminus U_{n-1}$. In which case, if $n > 1$, then $x \in U_{n} \setminus \ol{U_{n - 2}} = V_{n-1}$. If $n = 1$, then $x \in U_{2} = V_1$. Thus $\seq{V_n}$ is an open cover of $X$. In addition, for any $m, n \in \natp$ with $m \le n$, $V_m \cap V_n \ne \emptyset$ implies that $n - m < 2$, so $\seq{V_n}$ is locally finite. By (2) of \autoref{proposition:lch-paracompact}, $X$ is paracompact.
 \end{proof}
 
-

src/topology/main/support.tex

@@ -8,9 +8,20 @@
 
 \begin{definition}[Compactly Supported]
 \label{definition:compactly-supported}
-    Let $X$ be a topological space, $E$ be a TVS over $K \in \RC$, and $f \in C(X; E)$, then $f$ is \textbf{compactly supported} if $\supp{f}$ is compact. The set $C_c(X; E)$ is the vector space of all $E$-valued compactly supported functions on $X$.
+    Let $X$ be a topological space, $E$ be a TVS over $K \in \RC$, and $f \in C(X; E)$, then $f$ is \textbf{compactly supported} if $\supp{f}$ is compact. The set $C_c(X; E)$ is the space of $E$-valued compactly supported continuous functions on $X$.
 \end{definition}
 
+\begin{definition}[Non-negative Compactly Supported Functions]
+\label{definition:non-negative-compactly-supported}
+    Let $X$ be a topological space, then
+    \[
+    C_c^+(X) = \bracs{f \in C_c(X)|f \ge 0}
+    \]
+
+    is the space of \textbf{non-negative compactly supported continuous functions} on $X$. 
+\end{definition}
+
+
 \begin{definition}
 \label{definition:compactly-supported-01}
     Let $X$ be a topological space, $f \in C_c(X; [0, 1])$ and $U \subset X$ be open, then $f \prec U$ if $\supp{f} \subset U$.

src/topology/notation.tex

@@ -22,6 +22,7 @@
     % Function Spaces
     $\mathrm{supp}(f)$ & Support of $f$. & \autoref{definition:support} \\
     $C_c(X; E)$ & Compactly supported continuous functions $X \to E$. & \autoref{definition:compactly-supported} \\
+    $C_c^+(X)$ & Compactly supported continuous functions $X \to [0, \infty)$ & \autoref{definition:non-negative-compactly-supported}
     $f \prec U$ & $f \in C_c(X; [0,1])$ with $\mathrm{supp}(f) \subset U$. & \autoref{definition:compactly-supported-01} \\
     $C_0(X; E)$ & Continuous functions vanishing at infinity. & \autoref{definition:vanish-at-infinity} \\
     $BC(X; E)$ & Bounded continuous functions $X \to E$. & \autoref{definition:bounded-continuous-function-space} \\