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\section{Introduction}

\subsection{Some Questions}

\subsubsection*{Holes in Classical Theory of Analysis}

\begin{enumerate}[(1)]
  \item What is the ``volume'' of a subset of
      $\RR^d$? $d = 2$ we have ``area'', $d = 1$
      we have ``length'' (we know the length of
      intervals).
  \item Integration: Riemann Integral has holes.
    Let $\{f_n\}$ be a sequence of continuous
    functions on $[0, 1]$ such that $0 \le
    f_n(x) \le 1$ for all $x \in [0, 1]$,
    $f_n(x)$ is monotonically decreasing as $n
    \to \infty$, i.e. $f_n(x) \ge f_{n +
    1}(x)$ for all $x$. So, $\lim_{n \to
    \infty} f_n(x) = f(x)$ exists for all $x$.
    So $\lim_{n \to \infty} \int_0^1 f_n(x)
    \dd x$ exists. But $f$ is not
    \emph{necessarily} Riemann integrable. We
    want a new theory of integrals such that
    $f$ is integrable, and such that $\lim_{n
    \to \infty} \int_0^1 f_n(x) \dd x =
    \int_0^1 f(x) \dd x$.
  \item Let $L' = \ol{(\mathcal{C}[0, 1], \|
    \bullet \|_1)}$. If $f \in L'$, is $f$ Riemann
    integrable? ($\|f\|_1 = \int_{0^1} |f(x)| \dd
    x$). Will have to change the definition
    of integral. $L^2 = \text{a Hilbert space}$
    \to Fourier Analysis.
\end{enumerate}

\subsubsection*{Holes in Classical Theory of Probability}

\begin{enumerate}[(1)]
  \item Discrete probability has its limitations.
    \begin{itemize}
      \item Toss an unbiased coin 5 times. What is
        the probability of getting 3 heads? This
        is a question we know how to answer.
      \item Take an infinite sequence of coin
        tosses and an event $A$ that depends on
        that infinite sequence. How to define
        $P(A)$? (For example Strong Law of Large
        Numbers).
      \item How to draw a point uniformly at
        random from $[0, 1]$?
    \end{itemize}
    Probability needs \emph{axioms} to be made
    rigorous.
  \item Define expectation ($\EE$) for a random
    variable. Also would want the following: if
    $0 \le X_n \le 1$ and $X_n \downarrow X$, then
    $\EE X_n \to \EE X$.
\end{enumerate}

\subsection{Basic Definitions}

\begin{flashcard}[sigma-algebra]
\begin{definition*}[$\sigma$-algebra]
  Let $E$ be a set. A $\sigma$-algebra
  $\mathcal{E}$ is \cloze{a collection of subsets of $E$
  such that
  \begin{itemize}
    \item $\emptyset \in \mathcal{E}$
    \item if $A \in \mathcal{E}$, then $A^c \in
      \mathcal{E}$ ($A^c = E \setminus A$)
    \item if $(A_n : n \in \NN)$, $A_n \in
      \mathcal{E} \forall n$, then $\bigcup_n A_n
      \in \mathcal{E}$ too.
  \end{itemize}
  $(E, \mathcal{E})$ is called a \emph{measurable space}. 
  }
\end{definition*}
\end{flashcard}

\begin{example*}
  $\mathcal{E} = (\emptyset, E)$ or $\mathcal{E} =
  \mathcal{P}(E)$ (power set). Typically, we will
  deal with things somewhere between these
  extremes.
\end{example*}

\begin{remark*}
  Since $\bigcap_n A_n = \left( \bigcup_n A_n^c
  \right)^c$, any $\sigma$-algebra is stable
  under countable intersections. Also, if $a, B
  \in \mathcal{E}$, then $B \setminus A = B \cap
  A^c \in \mathcal{E}$.
\end{remark*}

\begin{flashcard}[measure-defn]
\begin{definition*}[Measure]
  A \emph{measure} $\mu$ on $(E, \mathcal{E}$) is
  a \cloze{non-negative function $\mu : \mathcal{E} \to
  [0, \infty]$ such that
  \begin{itemize}
    \item $\mu(\emptyset) = 0$
    \item For all sequences $A_n$, $n \in \NN$
      with $A_n \in \mathcal{E}$ and all $A_n$
      pairwise disjoint, we have countable
      additivity:
      \[ \mu \left( \bigcup_{n = 1}^\infty A_n
      \right) = \sum_{n = 1}^\infty \mu(A_n) \]
  \end{itemize}
  We call $(e, \mathcal{E}, \mu)$ a \emph{measure
  space}. 
  }
\end{definition*}
\end{flashcard}

\begin{remark*}
  Let $E$ be a countable set, with $\mathcal{E} =
  \mathcal{P}(E)$. Then $\forall A
  \subset E$, $\mu(A) = \mu \left( \bigcup_{x \in
  A} \{x\} \right) = \sum_{x \in A} \mu(\{x\}) =
  \sum_{x \in A} m(x)$, where we define $m : E \to
  [0, \infty]$ such that $m(x) = \mu(\{x\}$. We
  call such an $m$ a ``mass function'' (or pmf in
  discrete probability), and measures $\mu$ are in
  one-to-one correspondence with mass function
  $m$. Here $\mathcal{E} = \mathcal{P}(E)$ and
  this is the theory in elementary discrete
  probability (when $\mu(\{x\}) = 1$ for all $x
  \in E$, $\mu$ is called a counting measure, and
  here $\mu(A) = |A|$ for all $A \subset E$).
\end{remark*}
\vspace{-1em}

\noindent
For uncountable $E$ however, the story is not so
simple and $\mathcal{E} = \mathcal{P}(E)$ is
generally not feasible. Instead measures are
defined on $\sigma$-algebras ``\emph{generated}''
by a smaller class $\mathcal{A}$ of simple subsets of $E$.

\begin{flashcard}[sigma-algebra-generated-by-defn]
\begin{definition*}[Generated $\sigma$-algebra]
  \cloze{
  If $\mathcal{A}$ is any collection of subsets of
  $E$, we define
  \[ \sigma(\mathcal{A}) = \{A \subseteq E : A \in
  \mathcal{E} \forall \text{ $\sigma$-algebras }
  \mathcal{E} \supseteq \mathcal{A}\} =
  \bigcap_{\substack{\mathcal{E}
  \supseteq \mathcal{A} \\ \text{$\mathcal{E}$ a
  $\sigma$-algebra}}}
  \mathcal{E} \]
  We call this \emph{the $\sigma$-algebra
  generated by $\mathcal{A}$}. It is the smallest
  $\sigma$-algebra containing $\mathcal{A}$.
  }
\end{definition*}
\end{flashcard}
\vspace{-1em}

\noindent
Why is $\sigma(\mathcal{A})$ a $\sigma$-algebra?
Answer: Example Sheet 1 problem 1.

The class $\mathcal{A}$ will usually satisfy some
properties too.

\begin{flashcard}[measure-ring]
\begin{definition*}[Ring]
  \cloze{
  Let $E$ be a set and $\mathcal{A}$ a collection
  of subsets of $E$. $\mathcal{A}$ is called a
  \emph{ring} if
  \begin{itemize}
    \item $\emptyset \in \mathcal{A}$
    \item For all $A, B \in \mathcal{A}$, $A \cup
      B \in \mathcal{A}$ and $B \setminus A \in
      \mathcal{A}$.
  \end{itemize}
  }
\end{definition*}
\end{flashcard}

\begin{remark*}
  If $A, B \in \mathcal{A}$, $\mathcal{A}$ a ring,
  then
  \[ A \triangle B = (A \setminus B) \cup (B
  \setminus A) \in \mathcal{A} \]
  \[ A \cap B = (A \cup B) \setminus (A \triangle
  B) \in \mathcal{A} \]
\end{remark*}

\begin{flashcard}[measure-algebra]
\begin{definition*}[Algebra]
  $\mathcal{A}$ is called an \emph{algebra}
  if\cloze{
  \begin{itemize}
    \item $\emptyset \in \mathcal{A}$
    \item If $A, B \in \mathcal{A}$, then $A^c \in
      \mathcal{A}$ and $A \cup B \in \mathcal{A}$.
  \end{itemize}
  }
\end{definition*}
\end{flashcard}

\begin{remark*}
  If $\mathcal{A}$ an algebra and $A, B \in
  \mathcal{A}$, then $A \cap B \in \mathcal{A}$
  and $B \setminus A = B \cap A^c \in
  \mathcal{A}$. So an algebra is also a ring.
  $\{\emptyset\}$ is a ring, but not an algebra.
\end{remark*}
\vspace{-1em}

\noindent
The idea:
\begin{itemize}
  \item Define a set function on a suitable
    collection $\mathcal{A}$.
  \item Extend the set function to a measure on
    $\sigma(\mathcal{A})$ (Caratheodory's
    Extension theorem).
  \item Such an extension is unique (Dynkin's
    lemma).
\end{itemize}
