polyrectangle Polyrectangle 2005-02-18 08:48:02 2007-06-29 03:58:01 Definition Riemann multiple integral Riemann sums on polyrectangles compact rectangle % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \newcommand{\R}{\mathbb R} A \emph{polyrectangle} $P$ in $\R^n$ is a finite collection $P=\{R_1,\ldots,R_N\}$ of compact rectangles $R_i\subset \R^n$ with disjoint interior. A \emph{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 $a_j^i<b_j^i$ (these are also called \emph{$n$-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 $\R^n$. We can define the ($n$-dimensional) measure of $\cup P$ in a \PMlinkescapetext{simple} 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 \emph{upper} and \emph{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 \emph{refinements}. The family of rectangles $R_i$ which appear in the definition~\ref{defpoly} are called a \emph{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 $S$.