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\date{26 March 2018}
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\begin{document}

\begin{frame}{Axioms of Outer measure}
\underline{Axioms of Outer Measure} \\
Recall the power set $\mathscr{P}(X)$ of a set $X$ is the set of all subsets. We say that a function $m^* \colon \mathscr{P}(X) \rightarrow [0,\infty]$ (note: we allow $\infty$!) is an outer measure on $X$ provided it satisfies the following three axioms.
\begin{enumerate}[a.)]
\item The outer measure of the empty set is zero.
\item (\textbf{monotonicity}) If $A \subseteq B$, then $m^* A \leq m^* B$.
\item (\textbf{countable subadditivity}) If $A = \bigcup_{n=1}^{\infty} A_n$, then $m^* A \leq \displaystyle\sum_{n=1}^{\infty} m^* A_n$.
\end{enumerate}
\end{frame}

\begin{frame}{Measure}
From an outer measure, we may define a \textbf{measure} on the set of ``measurable sets". Let $\mathcal{M}$ denote the measurable sets with respect to some outer measure. Then we say that the map $m \colon \mathcal{M} \rightarrow [0,\infty]$ is a measure. It obeys the three ``measure axioms":
\begin{enumerate}[a.)]
\item The measure of the empty set is zero.
\item (\textbf{monotonicity}) If $A \subseteq B$, then $m A \leq m B$.
\item (\textbf{countable additivity}) If $A = \bigcup_{n=1}^{\infty} A_n$ and each $A_i \cap A_j = \emptyset$	, then $m A = \displaystyle\sum_{n=1}^{\infty} m A_n$.
\end{enumerate}
\end{frame}

\begin{frame}{Measurable functions}
\textbf{Definition}: Let $X \subset \mathbb{R}$. A function $f \colon X \rightarrow [-\infty,\infty]$ is called measurable if $X$ is measurable and for all $\alpha \in \mathbb{R}$, the set $\{x \colon f(x) \geq \alpha\}$ is measurable. \\
\textbf{Proposition}: If $c$ is a constant and $f,g$ are measurable functions, then $f+c, cf, f+g, g-f,$ and $fg$ are measurable. \\
\textbf{Proposition}: Continuous functions are measurable, step functions are measurable, restrictions of measurable functions to measurable domains are measurable.  \\
\end{frame}

\begin{frame}{Littlewood's 3 principles}
\begin{enumerate}[1.]
\item Every measurable set is nearly a finite union of intervals. 
\item Every measurable function is nearly continuous.
\item Every convergent sequence of measurable functions is nearly uniformly convergent.
\end{enumerate}

\end{frame}

\begin{frame}{Lebesgue integration -- ``simple functions"}
We saw earlier: Riemann and Darboux integration. Now we build up the measure theoretic integral (in turbo speed). 

Let $E \subset \mathbb{R}$ and define the characteristic function $\chi_E \colon \mathbb{R} \rightarrow \{0,1\}$ by
$$\chi_E(x) = \left\{ \begin{array}{ll}
1, & \quad x \in E \\ 
0, & \quad x \not\in E
\end{array} \right.$$
A linear combination of characteristic functions is called a simple function: 
$$\psi(x) = \displaystyle\sum_{i=1}^n c_k \chi_{E_i}(x).$$
\textbf{Quiz}: Let $E_1=[0,1], E_2=[2,3] \cup [4,5]$, and $E_3=[10,11]$. Draw a picture of the simple function $\psi \colon \mathbb{R} \rightarrow \mathbb{R}$ defined by $\psi(x)=\displaystyle\sum_{k=1}^3 k^2 \chi_{E_k}(x)$
\end{frame}

\begin{frame}{Integration of nonnegative simple functions}
If $m$ is a measure, and $\psi \colon X \rightarrow \mathbb{R}$ defined by $\psi(x)=\displaystyle\sum_{k=1}^n c_k \chi_{E_k}(x)$ is an non-negative simple function. If $E \subset X$ is measurable, then we define
$$\displaystyle\int_E \psi dm = \displaystyle\sum_{k=1}^n c_k m(E_k).$$

\textbf{Quiz}: Integrate the simple function from the previous slide.
\end{frame}

\begin{frame}{Integration of nonnegative $[0,\infty]$-valued functions}
Let $f \colon X \rightarrow [-\infty,\infty]$ and let $E \subset X$ be measurable. Consider the set of simple functions $\psi$ with the property that $0 \leq \psi \leq f$.  We define the integral of $f$ by
$$\displaystyle\int_E f \mathrm{d}m = \sup_{0 \leq \psi \leq f} \displaystyle\int_E \psi \mathrm{d}m$$
\end{frame}

\begin{frame}{Positive and negative parts} 
Let $f^+$ denote the positive part of a function $f$, i.e.
$$f^+(x) = \max \{f(x),0\}.$$
Similarly let $f^-$ be the negative part:
$$f^-(x) = \min \{-f(x),0\}.$$
Fact: if $f$ is measurable, then so are $f^+$ and $f^-$ and we have
$$f=f^+ - f^-.$$

\textbf{Quiz}: Let $f \colon \mathbb{R} \rightarrow \mathbb{R}$ be defined by $f(x)=x^3$. What is $f^+$? What is $f^-$?
\end{frame}

\begin{frame}{General Lebesgue integral}
To integrate a function $f \colon X \rightarrow [-\infty,\infty]$ we integrate its positive and negative parts:
$$\displaystyle\int_E f \mathrm{d}m = \displaystyle\int_E f^+ \mathrm{d}m - \displaystyle\int_E f^- \mathrm{d}m.$$
\end{frame}

\begin{frame}{Probability theory}
Sometimes in a measure space, the measure of the whole space is infinite (for example, the Lebesgue measure is $\infty$ when measuring $\mathbb{R}$ and the cardinality measure is $\infty$ when measuring any infinite set). \\
\vspace{0.1in}
Suppose we have a set $X$ and a measure $m \colon \mathcal{M} \rightarrow [0,\infty]$, where $\mathcal{M}$ is some collection of measurable sets. If $m(X)=1$, then we call this structure a probability measure space.
\end{frame}

\begin{frame}{Probability theory}
Consider the sample space $\Omega$. Let $E$ be any event and let $A$ and $B$ be events so that $A \subseteq B$. Let $C$ and $D$ be mutually exclusive events. Let $X$ be a ``random variable" (what is this??). 
\onslide<2->{\begin{tabular}{|l|l|}
\hline
Property from Probability Class & Property of Measure \\
\hline
probably of empty event: $P(\emptyset)=0$ & measure of empty set is $0$ \\
$P(A) \leq P(B)$ & monotone \\
$P(C \cup D)=P(C)+P(D)$ & countable additivity \\
\hline
\end{tabular} \\
These similarities are no coincidence!!}
\end{frame}
\iffalse
\begin{frame}{What is a random variable?}
Let $\Omega$ be a set and let $m$ be a measure on the set $\mathcal{M}$ of measurable subsets of $\Omega$ that makes $\Omega$ into a probability measure space.  \\
\vspace{0.1in}
To say ``let $X$ be a random variable" means that $X \colon \Omega \rightarrow \mathbb{R}$ is a measurable function. \\
\vspace{0.1in}
The cumulative distribution function of $X$, written $\mathrm{cdf}_X$, is a function $\mathrm{cdf}_X \colon \Omega \rightarrow [0,1]$ defined by
$$\mathrm{cdf}_X(t) = m(\{q \colon X(q) \leq t\}).$$
\end{frame}

\begin{frame}{Example}
\textbf{Die roll}: $\Omega=\{1,2,3,4,5,6\}$. Here $\mathcal{M}=\mathscr{P}(\Omega)$. We define our probability measure $m$ by $\dfrac{1}{6}$ of the cardinality measure. For instance, a random variable $X \colon \mathscr{P}(\Omega) \rightarrow [0,\infty)$ could encode the ``square of a die roll" and be defined as $X(t)=t^2$. The cdf of this random variable would be defined in the following way:
\begin{center}$\langle$board work$\rangle$\end{center}
\end{frame}

\begin{frame}{Probablity measure on a set of functions}
Let $\Omega=\{f_0,f_1,f_2,f_3,f_4\}$ where $f_i \colon [0,1] \rightarrow \mathbb{R}$ are defined by $f_i(t)=t^i$ for $i=0,1,2,3,4$. Let $\mathcal{M}=\mathscr{P}(\Omega)$. Let $E \subset \mathcal{M}$ and define a probability measure by
$$m(E) = \dfrac{60}{137} \displaystyle\int_0^1 \displaystyle\sum_{f_i \in E} f_i(t) \mathrm{d}t.$$

\end{frame}
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