# 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