Riemann multiple integral

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 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 $S^{*}(f,P)\geq S_{*}(f,P)$ . Also one has $S^{*}(f,P)\geq S_{*}(f,P^{\prime})$ when $P$ and $P^{\prime}$ are any two polyrectangles containing the support of $f$ . In fact one can always find a common refinement $P^{\prime\prime}$ of both $P$ and $P^{\prime}$ so that $S^{*}(f,P)\geq S^{*}(f,P^{\prime\prime})\geq S_{*}(f,P^{\prime\prime})\geq S_{% *}(f,P^{\prime})$ . 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 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 $\mathbb{R}^{n}$ (as defined above). Moreover we define the 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 volume of $D$ , and when $n=2$ the Peano Jordan measure of $D$ is called the 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 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 Riemann integral of $f$ on $D$ .

Title	Riemann multiple integral
Canonical name	RiemannMultipleIntegral
Date of creation	2013-03-22 15:03:34
Last modified on	2013-03-22 15:03:34
Owner	paolini (1187)
Last modified by	paolini (1187)
Numerical id	14
Author	paolini (1187)
Entry type	Definition
Classification	msc 26A42
Related topic	Polyrectangle
Related topic	RiemannIntegral
Related topic	Integral2
Related topic	AreaOfPlaneRegion
Related topic	DevelopableSurface
Related topic	VolumeAsIntegral
Related topic	AreaOfPolygon
Related topic	MoscowMathematicalPapyrus
Related topic	IntegralOverPlaneRegion
Defines	Riemann integrable
Defines	Peano Jordan
Defines	measurable
Defines	area
Defines	volume
Defines	Jordan content