PlanetMath: Riemann multiple integral

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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

$\displaystyle S^*(f) := \inf\{ S^*(f,P) \colon$

is a polyrectangle,

for every $x\in\mathbb{R}^n\setminus \cup P$ $\displaystyle \},$

$\displaystyle S_*(f) := \sup\{ S_*(f,P) \colon$

is a polyrectangle,

for every $x\in\mathbb{R}^n\setminus \cup P$ $\displaystyle \}.$

we say that

is Riemann-integrable on $\mathbb{R}^n$ and we define the Riemann integral of

$\displaystyle \int f(x)\, dx := S^*(f) = S_*(f).$

Clearly one has $S^*(f,P)\ge S_*(f,P)$ . Also one has $S^*(f,P)\ge S_*(f,P')$ when and are any two polyrectangles. In fact one can always find a common refinement of both and 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 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 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 $\mathbb{R}^n$ (as defined above). Moreover we define the Peano-Jordan measure of

$\displaystyle \mathbf{meas}(D) := \int \chi_D(x)\, dx.$

When

the Peano Jordan measure of

is called the volume of

, and when

the Peano Jordan measure of

is called the area of

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 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 measurable as defined before. In this case we denote with

$\displaystyle \int_D f(x)\, dx := \int \bar f(x)\, dx$

the Riemann integral of

"Riemann multiple integral" is owned by paolini.

(view preamble)

See Also: polyrectangle, Riemann integral, Lebesgue integral

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

Cross-references: measurable set, characteristic function, bounded set, measure, domains, integral, refinement, lower Riemann sums, polyrectangle, support, compact, bounded function, variables, functions, Riemann integral
There are 196 references to this entry.

This is version 9 of Riemann multiple integral, born on 2005-02-18, modified 2006-09-01.
Object id is 6778, canonical name is RiemannMultipleIntegral.
Accessed 4682 times total.

Classification:

AMS MSC:

26A42 (Real functions :: Functions of one variable :: Integrals of Riemann, Stieltjes and Lebesgue type)

Pending Errata and Addenda

None.[ View all 3 ]

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