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In mathematics, the Peano–Jordan measure (also known as the Jordan content) is an extension of the notion of size ( length,
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an ope ...
,
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). Th ...
) to shapes more complicated than, for example, a
triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...
,
disk Disk or disc may refer to: * Disk (mathematics), a geometric shape * Disk storage Music * Disc (band), an American experimental music band * ''Disk'' (album), a 1995 EP by Moby Other uses * Disk (functional analysis), a subset of a vector sp ...
, or parallelepiped. It turns out that for a set to have Jordan measure it should be
well-behaved In mathematics, when a mathematical phenomenon runs counter to some intuition, then the phenomenon is sometimes called pathological. On the other hand, if a phenomenon does not run counter to intuition, it is sometimes called well-behaved. Th ...
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, despite the fact that it is not a true
measure Measure may refer to: * Measurement, the assignment of a number to a characteristic of an object or event Law * Ballot measure, proposed legislation in the United States * Church of England Measure, legislation of the Church of England * Mea ...
in its modern definition, since Jordan-measurable sets do not form a
σ-algebra In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set ''X'' is a collection Σ of subsets of ''X'' that includes the empty subset, is closed under complement, and is closed under countable unions and countabl ...
. For example, singleton sets \_in \mathbb each have a Jordan measure of 0, while \mathbb\cap ,1/math>, a countable union of them, is not Jordan-measurable. For this reason, some authors prefer to use the term ''Jordan
content Content or contents may refer to: Media * Content (media), information or experience provided to audience or end-users by publishers or media producers ** Content industry, an umbrella term that encompasses companies owning and providing mas ...
.'' The Peano–Jordan measure is named after its originators, the French mathematician
Camille Jordan Marie Ennemond Camille Jordan (; 5 January 1838 – 22 January 1922) was a French mathematician, known both for his foundational work in group theory and for his influential ''Cours d'analyse''. Biography Jordan was born in Lyon and educated at ...
, and the Italian mathematician
Giuseppe Peano Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician and glottologist. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation. The sta ...
.


Jordan measure of "simple sets"

Consider the
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
R''n''. One starts by considering products of bounded
intervals Interval may refer to: Mathematics and physics * Interval (mathematics), a range of numbers ** Partially ordered set#Intervals, its generalization from numbers to arbitrary partially ordered sets * A statistical level of measurement * Interval e ...
:C= unions_of_rectangles, :_S=C_1\cup_C_2\cup_\cdots_\cup_C_k 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_set.html" ;"title="union_(set_theory).html" ;"title="_1, b_1)\times [a_2, b_2) \times \cdots \times [a_n, b_n) 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: :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 union (set theory)">unions of rectangles, : S=C_1\cup C_2\cup \cdots \cup C_k 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 set">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.


Extension to more complicated sets

Notice that a set which is a product of closed intervals, :[a_1, b_1]\times [a_2, b_2] \times \cdots \times [a_n, b_n] is not a simple set, and neither is a ball (mathematics), ball. Thus, so far the set of Jordan measurable sets is still very limited. The key step is then defining a bounded set to be ''Jordan measurable'' if it is "well-approximated" by simple sets, exactly in the same way as a function is
Riemann integrable In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Göt ...
if it is well-approximated by piecewise-constant functions. Formally, for a bounded set ''B'', define its ''inner Jordan measure'' as :m_*(B)=\sup_ m (S) and its ''outer measure'' as :m^*(B)=\inf_ m (S) where the
infimum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest lo ...
and supremum are taken over simple sets ''S''. The set ''B'' is said to be Jordan measurable if the inner measure of ''B'' equals the outer measure. The common value of the two measures is then simply called the Jordan measure of ''B''. It turns out that all rectangles (open or closed), as well as all balls, simplexes, etc., are Jordan measurable. Also, if one considers two continuous functions, the set of points between the graphs of those functions is Jordan measurable as long as that set is bounded and the common domain of the two functions is Jordan measurable. Any finite union and intersection of Jordan measurable sets is Jordan measurable, as well as the set difference of any two Jordan measurable sets. A
compact set In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i. ...
is not necessarily Jordan measurable. For example, the
fat Cantor set In nutrition, biology, and chemistry, fat usually means any ester of fatty acids, or a mixture of such compounds, most commonly those that occur in living beings or in food. The term often refers specifically to triglycerides (triple e ...
is not. Its inner Jordan measure vanishes, since its
complement A complement is something that completes something else. Complement may refer specifically to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Aggregate complementation, the separation of pitch-clas ...
is
dense Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
; however, its outer Jordan measure does not vanish, since it cannot be less than (in fact, is equal to) its Lebesgue measure. Also, a bounded
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
is not necessarily Jordan measurable. For example, the complement of the fat Cantor set (within the interval) is not. A bounded set is Jordan measurable if and only if its indicator function is Riemann-integrable, and the value of the integral is its Jordan measur

Equivalently, for a bounded set ''B'' the inner Jordan measure of ''B'' is the Lebesgue measure of the interior (topology), interior of ''B'' and the outer Jordan measure is the Lebesgue measure of the closure. From this it follows that a bounded set is Jordan measurable if and only if its
boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment * ''Boundaries'' (2016 film), a 2016 Canadian film * ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film *Boundary (cricket), the edge of the pla ...
has Lebesgue measure zero. (Or equivalently, if the boundary has Jordan measure zero; the equivalence holds due to compactness of the boundary.)


The Lebesgue measure

This last property greatly limits the types of sets which are Jordan measurable. For example, the set of
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rat ...
s contained in the interval ,1is then not Jordan measurable, as its boundary is ,1which is not of Jordan measure zero. Intuitively however, the set of rational numbers is a "small" set, as it is
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...
, and it should have "size" zero. That is indeed true, but only if one replaces the Jordan measure with the Lebesgue measure. The Lebesgue measure of a set is the same as its Jordan measure as long as that set has a Jordan measure. However, the Lebesgue measure is defined for a much wider class of sets, like the set of rational numbers in an interval mentioned earlier, and also for sets which may be unbounded or
fractals In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illus ...
. Also, the Lebesgue measure, unlike the Jordan measure, is a true
measure Measure may refer to: * Measurement, the assignment of a number to a characteristic of an object or event Law * Ballot measure, proposed legislation in the United States * Church of England Measure, legislation of the Church of England * Mea ...
, that is, any countable union of Lebesgue measurable sets is Lebesgue measurable, whereas countable unions of Jordan measurable sets need not be Jordan measurable.


References

* *


External links

* *{{springer , title= Jordan measure , id= J/j054350 , last= Terekhin , first= A.P. , author-link= Measures (measure theory)