A polyrectangle P in n is a finite collectionMathworldPlanetmath P={R1,,RN} of compact rectangles Rin with disjoint interior. A compact rectangle Ri is a Cartesian product of compactPlanetmathPlanetmath intervals: Ri=[a1i,b1i]××[ani,bni] where aji<bji (these are also called n-dimensional intervals).

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


It is a compact subset of n.

We can define the (n-dimensional) measure of P in a way. If R=[a1,b1]××[an,bn] is a rectangle we define the measure of R as


and define the measure of the polyrectangle P as:


Moreover if we are given a bounded function f:P we can define the upper and lower Riemann sums of f over P by


Polyrectangles are then used to define the Peano Jordan measure of subsets of n and to define Riemann multiple integrals. To achieve this, it is useful to introduce the so called refinementsPlanetmathPlanetmath. The family of rectangles Ri which appear in the definition LABEL:defpoly are called a partitionPlanetmathPlanetmath of P¯ in rectangles. It is clear that the set P can be represented by different polyrectangles. For example any rectangle R can be split in 2n 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 (P)(Q)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