Jordan Measure - Jordan Measure of "simple Sets"

Jordan Measure of "simple Sets"

Consider the Euclidean space Rn. One starts by considering products of bounded intervals

which are closed at the left end and open at the right end (half-open intervals is a technical choice; as we see below, one can use closed or open intervals if preferred). Such a set will be called a n-dimensional rectangle, or simply a rectangle. One defines the Jordan measure of such a rectangle to be the product of the lengths of the intervals:

Next, one considers simple sets, sometimes called polyrectangles, which are finite unions of rectangles,

for any k≥1. One cannot define the Jordan measure of S as simply the sum of the measures of the individual rectangles, because such a representation of S is far from unique, and there could be significant overlaps between the rectangles. Luckily, any such simple set S can be rewritten as a union of another finite family of rectangles, rectangles which this time are mutually disjoint, and then one defines the Jordan measure m(S) as the sum of measures of the disjoint rectangles. One can show that this definition of the Jordan measure of S is independent of the representation of S as a finite union of disjoint rectangles. It is in the "rewriting" step that the assumption of rectangles being made of half-open intervals is used.