Riemann multiple integral RiemannMultipleIntegral 2005-02-18 09:17:42 2007-07-18 05:47:50 Definition Riemann integrable Peano Jordan measurable area volume Jordan content % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions \usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \newcommand{\R}{\mathbb R} \newtheorem{theorem}{Theorem} \newtheorem{definition}{Definition} \theoremstyle{remark} \newtheorem{example}{Example} We are going to extend the concept of Riemann integral to functions of several variables. Let $f\colon\mathbb R^n \to\mathbb R$ be a bounded function with compact support. Recalling the definitions of polyrectangle and the definitions of upper and lower Riemann sums on polyrectangles, we define \[ S^*(f) := \inf\{ S^*(f,P) \colon \text{$P$ is a polyrectangle, $f(x)=0$ for every $x\in\mathbb R^n\setminus \cup P$}\}, \] \[ S_*(f) := \sup\{ S_*(f,P) \colon \text{$P$ is a polyrectangle, $f(x)=0$ for every $x\in\mathbb R^n\setminus \cup P$}\}. \] If $S^*(f)=S_*(f)$ we say that $f$ is \emph{Riemann-integrable} on $\mathbb R^n$ and we define the Riemann integral of $f$: \[ \int f(x)\, dx := S^*(f) = S_*(f). \] Clearly one has $S^*(f,P)\ge S_*(f,P)$. Also one has $S^*(f,P)\ge S_*(f,P')$ when $P$ and $P'$ are any two polyrectangles containing the support of $f$. In fact one can always find a common refinement $P''$ of both $P$ and $P'$ so that $S^*(f,P)\ge S^*(f,P'')\ge S_*(f,P'')\ge S_*(f,P')$. So, to prove that a function is Riemann-integrable it is enough to prove that for every $\epsilon>0$ there exists a polyrectangle $P$ such that $S^*(f,P)-S_*(f,P)<\epsilon$. Next we are going to define the integral on more general domains. As a byproduct we also define the measure of sets in $\mathbb R^n$. Let $D\subset \mathbb R^n$ be a bounded set. We say that $D$ is \emph{Riemann measurable} if the characteristic function \[ \chi_D(x):=\begin{cases} 1 &\text{if $x\in D$}\\ 0 &\text{otherwise}\end{cases} \] is Riemann measurable on $\R^n$ (as defined above). Moreover we define the \emph{Peano-Jordan measure} of $D$ as \[ \mathbf{meas}(D) := \int \chi_D(x)\, dx. \] When $n=3$ the Peano Jordan measure of $D$ is called the \emph{volume} of $D$, and when $n=2$ the Peano Jordan measure of $D$ is called the \emph{area} of $D$. Let now $D\subset \mathbb R^n$ be a Riemann measurable set and let $f\colon D\to \mathbb R$ be a bounded function. We say that $f$ is \emph{Riemann measurable} if the function $\bar f\colon\mathbb R^n\to\mathbb R$ \[ \bar f(x) :=\begin{cases} f(x)&\text{if $x\in D$}\\ 0&\text{otherwise} \end{cases} \] is Riemann integrable as defined before. In this case we denote with \[ \int_D f(x)\, dx := \int \bar f(x)\, dx \] the \emph{Riemann integral} of $f$ on $D$.