# polyrectangle

A polyrectangle $P$ 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 $R_{i}$ 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 $a_{j}^{i} (these are also called $n$-dimensional intervals).

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

 $\cup P:=\bigcup_{R\in P}R=R_{1}\cup\cdots\cup R_{N}.$

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

We can define the ($n$-dimensional) measure of $\cup P$ in a way. If $R=[a_{1},b_{1}]\times\cdots\times[a_{n},b_{n}]$ is a rectangle we define the measure of $R$ as

 $\mathrm{meas}(R):=(b_{1}-a_{1})\cdots(b_{n}-a_{n})$

and define the measure of the polyrectangle $P$ as:

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

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 $f$ over $\cup P$ by

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

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 $R_{i}$ which appear in the definition LABEL:defpoly 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 $R$ can be split in $2^{n}$ smaller rectangles by dividing in two parts each of the $n$ intervals defining $R$. We claim that given two polyrectangles $P$ and $Q$ there exists a polyrectangle $S$ such that $(\cup P)\cup(\cup Q)\subset\cup S$ and such that given any rectangle $R$ in $P$ or $Q$, $R$ is the union of rectangles in $S$.

Title polyrectangle Polyrectangle 2013-03-22 15:03:31 2013-03-22 15:03:31 paolini (1187) paolini (1187) 23 paolini (1187) Definition msc 26A42 RiemannMultipleIntegral Riemann sums on polyrectangles compact rectangle