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 { x } x ∈ R {\displaystyle \{x\}_{x\in \mathbb {R} }} in R {\displaystyle \mathbb {R} } each have a Jordan measure of 0, while Q ∩ [ 0 , 1 ] {\displaystyle \mathbb {Q} \cap [0,1]} , a countable union of them, is not Jordan-measurable. For this reason, some authors prefer to use the term Jordan content. The Peano–Jordan measure is named after its originators, the French mathematician Camille Jordan, and the Italian mathematician Giuseppe Peano.
Jordan measure of "simple sets"
Consider Euclidean space R n . {\displaystyle \mathbb {R} ^{n}.} Jordan measure is first defined on Cartesian products of bounded half-open intervals
C = [ a 1 , b 1 ) × [ a 2 , b 2 ) × ⋯ × [ a n , b n ) {\displaystyle C=[a_{1},b_{1})\times [a_{2},b_{2})\times \cdots \times [a_{n},b_{n})}
that are closed at the left and open at the right with all endpoints a i {\displaystyle a_{i}} and b i {\displaystyle b_{i}} 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 n {\displaystyle n} -dimensional rectangle, or simply a rectangle. The Jordan measure of such a rectangle is defined to be the product of the lengths of the intervals:
m ( C ) = ( b 1 − a 1 ) ( b 2 − a 2 ) ⋯ ( b n − a n ) . {\displaystyle m(C)=(b_{1}-a_{1})(b_{2}-a_{2})\cdots (b_{n}-a_{n}).}
Next, one considers simple sets, sometimes called polyrectangles, which are finite unions of rectangles,
S = C 1 ∪ C 2 ∪ ⋯ ∪ C k {\displaystyle S=C_{1}\cup C_{2}\cup \cdots \cup C_{k}}
for any k ≥ 1. {\displaystyle k\geq 1.}
One cannot define the Jordan measure of S {\displaystyle S} as simply the sum of the measures of the individual rectangles, because such a representation of S {\displaystyle S} is far from unique, and there could be significant overlaps between the rectangles. Luckily, any such simple set S {\displaystyle 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 ) {\displaystyle m(S)} as the sum of measures of the disjoint rectangles. One can show that this definition of the Jordan measure of S {\displaystyle S} is independent of the representation of S {\displaystyle 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.
Extension to more complicated sets
Notice that a set which is a product of closed intervals,
[ a 1 , b 1 ] × [ a 2 , b 2 ] × ⋯ × [ a n , b n ] {\displaystyle [a_{1},b_{1}]\times [a_{2},b_{2}]\times \cdots \times [a_{n},b_{n}]}
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