polyrectangle

A polyrectangle $P$ in ${ℝ}^n$ is a finite collection $P=\{R_1,\mathrm{\dots },R_N\}$ of compact rectangles $R_i\subset {ℝ}^n$ 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 ${ℝ}^n$.

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 $f:\cup P\to ℝ$ 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 ${ℝ}^n$ 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