updates?

chapters/chap0.tex

@@ -361,7 +361,24 @@ First, we define the Hilbert space in case one did not make the step from the li
     A Hilbert space is a complete inner product space.
 \end{defn}
 
-That is, a vector space equipped with an inner product that is complete (every Cauchy sequence converges to a limit).
+That is, a vector space equipped with an inner product, with the induced metric defined by the norm of the inner product, we have a metric space, which is complete. Reminds that complete mean that every Cauchy sequence, the sequence such that for any $\epsilon>0$, there exists an $N$ such that for all $m,n\geq N$, we have $|x_m-x_n|<\epsilon$, converges to a limit.
+
+As a side note we will use later, we also defined the Borel measure on a space, here we use the following definition specialized for the space (manifolds) we are interested in. 
+
+\begin{defn}
+    \label{defn:Borel_measure}
+    Borel measure:
+    
+    Let $X$ be a topological space, then a Borel measure $\mu:\mathscr{B}(X)\to [0,\infty]$ on $X$ is a measure on the Borel $\sigma$-algebra of $X$ $\mathscr{B}(X)$ satisfying the following properties:
+
+    \begin{enumerate}
+        \item $X \in \mathscr{B}$.
+        \item Close under complement: If $A\subseteq X$, then $\mu(A^c)=\mu(X)-\mu(A)$
+        \item Close under countable unions; If $E_1,E_2,\cdots$ are disjoint sets, then $\mu(\bigcup_{i=1}^\infty E_i)=\sum_{i=1}^\infty \mu(E_i)$
+    \end{enumerate}
+\end{defn}
+
+In later sections, we will use Lebesgue measure, and Haar measure for various circumstances, their detailed definition may be introduced in later sections.
 
 \begin{examples}
 
@@ -538,7 +555,7 @@ The counterpart of the random variable in the non-commutative probability theory
     \label{defn:observable}
     Observable:
 
-    Let $\mathscr{B}(\mathbb{R})$ be the set of all Borel sets on $\mathbb{R}$.
+    Let $\mathcal{B}(\mathbb{R})$ be the set of all Borel sets on $\mathbb{R}$.
 
     An (real-valued) observable (random variable) on the Hilbert space $\mathscr{H}$, denoted by $A$, is a projection-valued map (measure) $P_A:\mathscr{B}(\mathbb{R})\to\mathscr{P}(\mathscr{H})$.
 
@@ -721,7 +738,7 @@ $$
 \rho = \ket{\psi}\bra{\psi}.
 $$
 
-The probability that a measurement associated with a PVM $P$ yields an outcome in a Borel set $A$ is
+The probability that a measurement associated with a PVM $P$ yields an outcome in a Borel set $A\in \mathcal{B}$ is
 $$
 \mathbb P(A) = \operatorname{Tr}(\rho\, P(A)).
 $$
@@ -779,7 +796,7 @@ Here is Table~\ref{tab:analog_of_classical_probability_theory_and_non_commutativ
             \hline
             Sample space $\Omega$, cardinality $\vert\Omega\vert=n$, example: $\Omega=\{0,1\}$                                                                  & Complex Hilbert space $\mathscr{H}$, dimension $\dim\mathscr{H}=n$, example: $\mathscr{H}=\mathbb{C}^2$                                                   \\
             \hline
-            Common algebra of $\mathbb{C}$ valued functions                                                                                                     & Algebra of bounded operators $\mathscr{B}(\mathscr{H})$                                                                                                   \\
+            Common algebra of $\mathbb{C}$ valued functions                                                                                                     & Algebra of bounded operators $\mathcal{B}(\mathscr{H})$                                                                                                   \\
             \hline
             $f\mapsto \bar{f}$ complex conjugation                                                                                                              & $P\mapsto P^*$ adjoint                                                                                                                                    \\
             \hline
@@ -816,7 +833,7 @@ Here is Table~\ref{tab:analog_of_classical_probability_theory_and_non_commutativ
     \vspace{0.5cm}
 \end{table}
 
-\subsection{Quantum physics and terminologies}
+\section{Quantum physics and terminologies}
 
 In this section, we will introduce some terminologies and theorems used in quantum physics that are relevant to our study. Assuming no prior knowledge of quantum physics, we will provide brief definitions and explanations for each term.