\documentclass[pdftex,12pt,a4paper]{article}
\usepackage[usenames,dvipsnames]{color}
\usepackage[margin=1in]{geometry}
\usepackage[pdftex]{graphicx}
\usepackage[T1]{fontenc}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{verbatim}
\usepackage{mathpazo}
\usepackage{xfrac}
\usepackage{hyperref}

\setlength{\parindent}{0pt}

\newcommand{\block}{\mathbb}
\newcommand{\script}{\mathcal}
\newcommand{\fancy}{\mathfrak}
\newcommand{\C}{\block{C}}
\newcommand{\R}{\block{R}}
\newcommand{\Z}{\block{Z}}
\newcommand{\Q}{\block{Q}}
\newcommand{\N}{\block{N}}
\newcommand{\I}{^{-1}}
\newcommand{\set}[2]{\{#1|#2\}}
\newcommand{\topic}[1]{\noindent{\textbf{#1}}}
\newcommand{\bij}{\longleftrightarrow}
\newcommand{\bslash}{\setminus}
\newcommand{\cl}[1]{\overline{#1}}
\newcommand{\seq}{\subseteq}
\newcommand{\ds}{\displaystyle}
\newcommand{\Wlog}{Without loss of generality }
\newcommand{\rp}{$(\Rightarrow)$ }
\newcommand{\lp}{$(\Leftarrow)$ }
\renewcommand{\qedsymbol}{\tiny$\blacksquare$}
\newcommand{\cbox}[2]{\fcolorbox{#1}{white}{#2}}

\newtheorem{thm}{Theorem}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem{lem}{Lemma}

\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{ex}{Example}
\newtheorem{nex}{Non-Example}

\theoremstyle{remark}
\newtheorem{rec}{Recall}
\newtheorem{rem}{Remark}
\newtheorem{note}{Note}
\newtheorem{notate}{Notation}
\newtheorem{idea}{Idea}
\newtheorem{question}{Question}

\DeclareMathOperator{\mesh}{mesh}

\begin{document}

%%%%%%%%%%%%%
%\setlength{\topmargin}{-.9in}
\newcommand{\HRule}{\rule{\linewidth}{0.4mm}}
\begin{center}
\HRule \\
\textsc{Julius Elinson} \hfill \textsc{\today}\\[.1cm]
\textsc{\Large{Math 131 Notes}}\\[-.1cm] % Title
\HRule \\[.4cm]
\end{center}
%%%%%%%%%%%%%
\textsc{Disclaimer: }\small{Please feel free to use the pdf notes and/or the provided source code. However, the accuracy of these notes is in no way guaranteed. Furthermore, please do not use these notes instead of going to class, except in extenuating circumstances. Honor code applies.
Feel free to send any and all feedback, suggestions, or corrections to \href{mailto:jelinson@g.hmc.edu}{jelinson@hmc.edu}. }\\
\begin{defn}
 A partition $P$ of $[a,b]\seq\R$ is a set $P=\{x_0,\dots,x_n\}$ such that $a=x_0\leq x_1\leq \dots\leq x_{n-1}\leq x_n=b$. Notice that this is monotonically increasing but not necessarily strictly increasing. We denote $\Delta x_i=x_i-x_{i-1}$. The \emph{mesh} (or \emph{norm}) is $\mesh(p)=\max\{\Delta x_i|i=1,\dots,n\}$.
\end{defn}
\begin{defn}
 Another partition $Q=\{y_0,\dots,y_m\}$ on $[a,b]$ is a refinement of $P$ is $P\seq Q$.
\end{defn}
\begin{defn}
 Let $f$ be bounded on $[a,b]$. Let $P$ be a partition. Define
\begin{align*}
m_i&=\inf\{f(x)|x\in[x_{i-1},x_i]\}, \\
M_i&=\sup\{f(x)|x\in[x_{i-1},x_i]\}.
\end{align*}
The \emph{upper Darboux sum} and \emph{lower Darboux sum} of $f$ on $[a,b]$ with respect to $P$ are
\begin{align*}
 U(P,f)&=\sum_{i=1}^nM_i\Delta x_i,\\
 L(P,f)&=\sum_{i=1}^nm_i\Delta x_i.
\end{align*}
\end{defn}
\begin{defn}
 The \emph{upper and lower Darboux integrals} of $f$ over $[a,b]$ are 
\begin{align*}
 \overline{\int}_{\,a}^{\,b} f(x) dx &= \inf\{U(P,f)|\text{partitions $P$ on [a,b]}\},\\
  \underline{\int}_{\,a}^{\,b} f(x) dx &= \sup\{U(P,f)|\text{partitions $P$ on [a,b]}\}.
\end{align*}
\end{defn}
\begin{defn}
 If $\overline{\int}_{\,a}^{\,b} f(x) dx= \underline{\int}_{\,a}^{\,b} f(x) dx$ we say $f$ is Darboux integrable and we write $\int_a^bf(x)dx$.
\end{defn}
\begin{rem}[Reimann Sum]
 Take a single point in each partition and use the function value at that point to compute the approximate area.
\end{rem}
\begin{defn}
 A \emph{Riemann sum} of $f$ over $[a,b]$ with respect to $P$ is 
\begin{align*}
 R(P,f)=\sum_{i=1}^nf(c_i)\Delta x_i,
\end{align*}
with $c_i\in[x_{i-1},x_i]$.
\end{defn}
\begin{defn}
 For $s\in\R$, we say the \emph{Reimann integral} of $f$ over $[a,b]$ is equal to $s$ if for all $\epsilon>0$ there exists $\delta >0$ such that $\mesh(p)<\delta\iff|R(P,f)-s|<\epsilon$. In this case, we write $\int_a^bf(x)dx=s$.
\end{defn}
\begin{thm}
 Let $f$ be bounded on $[a,b]$. Then $f$ is Darboux integrable if and only if $f$ is Riemann integrable.
\end{thm}
\begin{note}
 Let $\fancy{R}$ denote the set of all Riemann integrable function.
\end{note}
\begin{defn}
 Let $f$ be bounded on $[a,b]$. Let $\alpha$ be monotonically increasing on $[a,b]$. Given a partition $P$, we write $\Delta \alpha_i=\alpha(x_i)-\alpha(x_{i-1})\geq0$. Then the \emph{upper and lower Riemann-Stieltjes} sums of $f$ over $[a,b]$ with respect to $P$ and $\alpha$ are:
\begin{align*}
 U(P,f,\alpha)&=\sum_{i=1}^nM_i\Delta \alpha_i,\\
 L(P,f,\alpha)&=\sum_{i=1}^nm_i\Delta \alpha_i.
\end{align*}
\end{defn}
\begin{defn}
 The upper and lower Riemann-Stieltjes integrals of $P$ over $[a,b]$ are:
\begin{align*}
 \overline{\int}_{\,a}^{\,b} f d\alpha&=\inf\{U(P,f,\alpha)|\text{partitions $P$ on [a,b]}\},\\
 \underline{\int}_{\,a}^{\,b} f d\alpha&=\inf\{L(P,f,\alpha)|\text{partitions $P$ on [a,b]}\}.
\end{align*}
If $\overline{\int}_{\,a}^{\,b} f d\alpha= \underline{\int}_{\,a}^{\,b} f d\alpha$, then we say $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ over $[a,b]$ and we write $\int_a^bfd\alpha$.
\end{defn}
\begin{note}
 The set of all Riemann-Stieltjes integrable functions with respect to $\alpha$ is denoted $\fancy{R}(\alpha)$.
\end{note}
\begin{ex}
 If $\alpha(x)=0$, then $\int_a^bf(x)dx=0$.
\end{ex}
\begin{ex}
 If $\alpha(x)=x$, then $\int_a^bfd\alpha=\int_a^bf(x)dx$.
\end{ex}
\begin{rem}
 We can regard the auxiliary function $\alpha$ as weighting of each partition.
\end{rem}
\begin{thm}
 If $Q$ is a refinement of $P$, then"
\begin{align*}
 L(P,f,\alpha)&\leq L(Q,f,\alpha),\\
 U(Q,f,\alpha)&\leq U(P,f,\alpha).
\end{align*}
\end{thm}
\begin{proof}
 Suppose $Q=P\cup{x^*}$ with $x^*(x_{i-1},x_i)$. Let:
\begin{align*}
 w_1&=\inf\{f(x)|x\in[x_{i-1},x^*]\},\\
 w_2&=\inf\{f(x)|x\in[x^*,x_i]\}.
\end{align*}
Notice that $w_1,w_2\geq m_i$. Thus:
\begin{align*}
 L(Q,f,\alpha)-L(P,f,\alpha) &= w_1[\alpha(x^*)-\alpha(x_{i-1})+w_2[\alpha(x_i)-\alpha(x^*)-m_i\Delta\alpha_i\\
 &=(w_1-m_i)[\alpha(x^*)-\alpha(x_{i-1})]+(w_2-m_i)[\alpha(x_i)-\alpha(x^*)]\\
 &\geq 0.
\end{align*}
The proof is completed by iterating this if $Q$ has more than one more point than $P$.
\end{proof}
\end{document}