A polyrectangle in is a finite collection of compact rectangles with disjoint interior. A compact rectangle is a Cartesian product of compact intervals: where (these are also called -dimensional intervals).
The union of the compact rectangles of a polyrectangle is denoted by
It is a compact subset of .
We can define the (-dimensional) measure of in a way. If is a rectangle we define the measure of as
and define the measure of the polyrectangle as:
Polyrectangles are then used to define the Peano Jordan measure of subsets of and to define Riemann multiple integrals. To achieve this, it is useful to introduce the so called refinements. The family of rectangles which appear in the definition LABEL:defpoly are called a partition of in rectangles. It is clear that the set can be represented by different polyrectangles. For example any rectangle can be split in smaller rectangles by dividing in two parts each of the intervals defining . We claim that given two polyrectangles and there exists a polyrectangle such that and such that given any rectangle in or , is the union of rectangles in .
|Date of creation||2013-03-22 15:03:31|
|Last modified on||2013-03-22 15:03:31|
|Last modified by||paolini (1187)|
|Defines||Riemann sums on polyrectangles|