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Concept
Jordan measure
Summary
In mathematics, the Peano–Jordan measure (also known as the Jordan content) is an extension of the notion of size (length, area, volume) to shapes more complicated than, for example, a triangle, disk, or parallelepiped.
It turns out that for a set to have Jordan measure it should be well-behaved in a certain restrictive sense. For this reason, it is now more common to work with the Lebesgue measure, which is an extension of the Jordan measure to a larger class of sets. Historically speaking, the Jordan measure came first, towards the end of the nineteenth century. For historical reasons, the term Jordan measure is now well-established for this set function, despite the fact that it is not a true measure in its modern definition, since Jordan-measurable sets do not form a σ-algebra. For example, singleton sets in each have a Jordan measure of 0, while , a countable union of them, is not Jordan-measurable. For this reason, some authors prefer to use the term .
The Peano–Jordan measure is named after its originators, the French mathematician Camille Jordan, and the Italian mathematician Giuseppe Peano.
Consider Euclidean space Jordan measure is first defined on Cartesian products of bounded half-open intervals
that are closed at the left and open at the right with all endpoints and finite real numbers (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 , or simply a . The of such a rectangle is defined to be the product of the lengths of the intervals:
Next, one considers , sometimes called , which are finite unions of rectangles,
for any
One cannot define the Jordan measure of as simply the sum of the measures of the individual rectangles, because such a representation of is far from unique, and there could be significant overlaps between the rectangles.
Luckily, any such simple set 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 as the sum of measures of the disjoint rectangles.
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