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 containing the support of . 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 $\bar f\colon\mathbb{R}^n\to\mathbb{R}$