# polyrectangle

A *polyrectangle* $P$ in ${\mathbb{R}}^{n}$ is a finite collection^{} $P=\{{R}_{1},\mathrm{\dots},{R}_{N}\}$ of compact rectangles ${R}_{i}\subset {\mathbb{R}}^{n}$ with disjoint interior.
A *compact rectangle* ${R}_{i}$ is a Cartesian product of compact^{} intervals: ${R}_{i}=[{a}_{1}^{i},{b}_{1}^{i}]\times \mathrm{\cdots}\times [{a}_{n}^{i},{b}_{n}^{i}]$ where $$ (these are also called *$n$-dimensional intervals*).

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

$$\cup P:=\bigcup _{R\in P}R={R}_{1}\cup \mathrm{\cdots}\cup {R}_{N}.$$ |

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

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

$$\mathrm{meas}(R):=({b}_{1}-{a}_{1})\mathrm{\cdots}({b}_{n}-{a}_{n})$$ |

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 $f:\cup P\to \mathbb{R}$ we can define the *upper* and *lower Riemann sums* of $f$ over $\cup P$ by

$${S}^{*}(f,P):=\sum _{R\in P}\mathrm{meas}(R)\underset{x\in R}{sup}f(x),{S}_{*}(f,P):=\sum _{R\in P}\mathrm{meas}(R)\underset{x\in R}{inf}f(x).$$ |

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 |
---|---|

Canonical name | Polyrectangle |

Date of creation | 2013-03-22 15:03:31 |

Last modified on | 2013-03-22 15:03:31 |

Owner | paolini (1187) |

Last modified by | paolini (1187) |

Numerical id | 23 |

Author | paolini (1187) |

Entry type | Definition |

Classification | msc 26A42 |

Related topic | RiemannMultipleIntegral |

Defines | Riemann sums on polyrectangles |

Defines | compact rectangle |