PlanetMath: polyrectangle

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polyrectangle

(Definition)

A polyrectangle in $\mathbb{R}^n$ is a finite collection $P=\{R_1,\ldots,R_N\}$ of compact rectangles $R_i\subset \mathbb{R}^n$ with disjoint interior. A compact rectangle is a Cartesian product of compact intervals: $R_i=[a_1^i,b_1^i]\times \cdots \times [a_n^i,b_n^i]$ where (these are also called -dimensional intervals).

The union of the compact rectangles of a polyrectangle is denoted by

$\displaystyle \cup P := \bigcup_{R\in P} R = R_1 \cup \cdots \cup R_N.$

It is a compact subset of $\mathbb{R}^n$ .

We can define the (-dimensional) measure of $\cup P$ in a simple way. If $R=[a_1,b_1]\times \cdots \times [a_n,b_n]$ is a rectangle we define the measure of as

$\displaystyle \mathrm{meas}(R) := (b_1-a_1)\cdots (b_n-a_n)$

and define the measure of the polyrectangle

as:

$\displaystyle \mathrm{meas}(P) := \sum_{R\in P} \mathrm{meas}(R).$

Moreover if we are given a bounded function $f\colon \cup P\to\mathbb{R}$ we can define the upper and lower Riemann sums of over $\cup P$ by

$\displaystyle S^*(f,P) := \sum_{R\in P} \mathrm{meas}(R)\sup_{x\in R} f(x),\qquad S_*(f,P) := \sum_{R\in P} \mathrm{meas}(R)\inf_{x\in R} f(x).$

Polyrectangles are then used to define the Peano Jordan measure of subsets of $\mathbb{R}^n$ and to define Riemann multiple integrals. To achieve this, it is useful to introduce the so called refinements. The family of rectangles which appear in the definition are called a partition of $\overline{\cup P}$ in rectangles. It is clear that the set $\cup P$ can be represented by different polyrectangles. For example any rectangle can be split in smaller rectangles by dividing in two parts each of the intervals defining . We claim that given two polyrectangles and there exists a polyrectangle such that $(\cup P)\cup (\cup Q) \subset \cup S$ and such that given any rectangle in or , is the union of rectangles in .

"polyrectangle" is owned by paolini.

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