Added more facts about topological groups.

src/measure/index.tex

@@ -8,4 +8,5 @@
 \input{./measurable-maps/index.tex}
 \input{./lebesgue-integral/index.tex}
 \input{./bochner-integral/index.tex}
+\input{./lcg/index.tex}
 \input{./notation.tex}

src/measure/lcg/index.tex

@@ -0,0 +1,4 @@
+\chapter{Integration on Locally Compact Groups}
+\label{chap:locally-compact-integration}
+
+\input{./lcg.tex}

src/measure/lcg/lcg.tex

@@ -0,0 +1,9 @@
+\section{Locally Compact Groups}
+\label{section:lcg}
+
+\begin{definition}[Locally Compact Group]
+\label{definition:lcg}
+    Let $G$ be a topological group, then $G$ is \textbf{locally compact} if $G$ is a LCH space. 
+\end{definition}
+
+

src/topology/groups/definition.tex

@@ -145,4 +145,43 @@
 \end{proposition}
 % Obvious so proof omitted.
 
+\begin{proposition}
+\label{proposition:tg-subgroup-closure}
+    Let $G$ be a topological group and $H \subset G$ be a subgroup, then $\ol H$ is also a subgroup.
+\end{proposition}
+\begin{proof}
+    By \autoref{proposition:closure-of-image}, for each $g \in G$, $g\ol H \subset \ol{gH} \subset \ol H$. Similarly, $\ol{H}^{-1} \subset \ol{H^{-1}} \subset \ol H$. 
+\end{proof}
+
+\begin{definition}[Left and Right Translations]
+\label{definition:tg-translations}
+    Let $G$ be a group, $g \in G$, and $Y$ be a set, then
+    \[
+    L_g: Y^G \to Y^G \quad (L_gf)(h) = f(g^{-1}h)
+    \]
+
+    is the \textbf{left translation map} by $g$, and
+    \[
+    R_g: Y^G \to Y^G \quad (R_gf)(h) = f(hg)
+    \]
+
+    is the \textbf{right translation map} by $g$. 
+\end{definition}
+
+\begin{proposition}
+\label{proposition:tg-translation-action}
+    Let $G$ be a group, $Y$ be a set, and $x, y \in G$, then:
+    \begin{enumerate}
+        \item $L_{xy} = L_xL_y$. 
+        \item $R_{xy} = R_xR_y$. 
+    \end{enumerate}
+\end{proposition}
+\begin{proof}
+    (1): For any $f \in Y^G$ and $z \in G$, 
+    \[
+    L_{xy}f(z) = f((xy)^{-1}z) = f(y^{-1}x^{-1}z) = L_yf(x^{-1}z) = L_xL_yf(z)
+    \]
+\end{proof}
+
+
 

src/topology/groups/index.tex

@@ -2,4 +2,5 @@
 \label{chap:topological-group}
 
 \input{./definition.tex}
+\input{./quotients.tex}
 

src/topology/groups/quotients.tex

@@ -0,0 +1,28 @@
+\section{Coset Spaces and Factor Groups}
+\label{section:tg-quotient}
+
+\begin{definition}[Coset Space]
+\label{definition:coset-space}
+    Let $G$ be a topological group and $H$ be a subgroup of $G$, then the set $\bracs{gH|g \in G}$ is the \textbf{coset space} of $G$ with respect to $H$, equipped with the quotient topology. 
+\end{definition}
+
+\begin{proposition}
+\label{proposition:coset-space-gymnastics}
+    Let $G$ be a topological group and $H$ be a subgroup of $G$, then:
+    \begin{enumerate}
+        \item The canonical map $\pi: G \to G/H$ is open. 
+        \item The coset space $G/H$ is Hausdorff if and only if $H$ is closed. 
+        \item The coset space $G/H$ is discrete if and only if $H$ is open. 
+    \end{enumerate}
+\end{proposition}
+\begin{proof}
+    (1): For each $U \subset G$, $\pi(U) = \pi(UH) = \pi^{-1}[\pi(U)]$. 
+
+    (2): If $H$ is closed, then for any $g \in G \setminus H$, there exists $U \in \cn_G(1)$ such that $Ug \cap UH = \emptyset$, so $\pi(Ug) \cap \pi(UH) = \emptyset$. 
+
+    On the other hand, if $G/H$ is Hausdorff, then $\ol H = \bigcup_{U \in \cn_G(1)}UH = H$, and $H$ is closed. 
+\end{proof}
+
+
+
+