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Riemann multiple integral (Definition)

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 $S^*(f,P)\ge S_*(f,P)$ . 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 Riemann measurable if the characteristic function

$\displaystyle \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 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 $\bar f\colon\mathbb R^n\to\mathbb R$

$\displaystyle \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 $D$ .




"Riemann multiple integral" is owned by paolini.
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See Also: polyrectangle, Riemann integral, Lebesgue integral, area of plane region, developable surface, volume as integral, area of polygon, Moscow Mathematical Papyrus, integral over plane region

Also defines:  Riemann integrable, Peano Jordan, measurable, area, volume, Jordan content

Attachments:
polyrectangle (Definition) by paolini
continuous functions of several variables are Riemann summable (Theorem) by paolini
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Cross-references: measurable set, characteristic function, Riemann, bounded set, measure, domains, integral, common refinement, lower Riemann sums, polyrectangle, definitions, support, compact, bounded function, variables, functions, Riemann integral
There are 171 references to this entry.

This is version 11 of Riemann multiple integral, born on 2005-02-18, modified 2007-07-18.
Object id is 6778, canonical name is RiemannMultipleIntegral.
Accessed 18336 times total.

Classification:
AMS MSC26A42 (Real functions :: Functions of one variable :: Integrals of Riemann, Stieltjes and Lebesgue type)

Pending Errata and Addenda
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