Jordan measure
The Jordan measure, also called Jordan content, is a finitely additive notion of size for bounded subsets of Euclidean space. It formalizes the geometric approximation of a set by finite unions of rectangular boxes. The construction agrees with ordinary length, area, and volume on the bounded regions customarily encountered in elementary geometry, but it applies to fewer sets than the Lebesgue measure.
A bounded set is Jordan measurable precisely when its inner and outer approximations can be made arbitrarily close using only finitely many boxes. An equivalent characterization states that the boundary of the set must have Jordan measure zero. This boundary criterion connects Jordan measure directly with Riemann integration, since the indicator function of a bounded set is Riemann integrable exactly when the set is Jordan measurable.
Elementary sets and volume
A bounded rectangle in (\mathbb{R}^n) is a Cartesian product
[ R=\prod_{i=1}^{n}[a_i,b_i], ]
where (a_i\leq b_i). Its (n)-dimensional volume is
[ \operatorname{vol}(R)=\prod_{i=1}^{n}(b_i-a_i). ]
The choice of open, closed, or half-open endpoints does not affect this volume. Half-open boxes are often used when constructing finite partitions because distinct boxes can then be made disjoint without assigning a positive volume to their shared faces.
An elementary set is a finite union of bounded rectangles. If an elementary set (E) is represented as a finite union of pairwise disjoint rectangles (R_1,\ldots,R_m), its elementary volume is
[ \operatorname{vol}(E)=\sum_{k=1}^{m}\operatorname{vol}(R_k). ]
This value is independent of the selected disjoint decomposition. Any two finite rectangular decompositions possess a common refinement, and finite additivity gives the same total volume after both decompositions are subdivided along their combined coordinate hyperplanes.
Elementary volume is monotone and finitely additive. Thus, if (E\subseteq F), then (\operatorname{vol}(E)\leq\operatorname{vol}(F)). If (E) and (F) are disjoint elementary sets, then
[ \operatorname{vol}(E\cup F)
\operatorname{vol}(E)+\operatorname{vol}(F). ]
These properties provide the finite geometric foundation from which Jordan inner and outer content are defined.
Inner and outer content
Let (A\subset\mathbb{R}^n) be bounded. Its Jordan inner content is
[ m_*(A)
\sup\left{ \operatorname{vol}(E): E\subseteq A,\ E\text{ elementary} \right}, ]
while its Jordan outer content is
[ m^*(A)
\inf\left{ \operatorname{vol}(E): A\subseteq E,\ E\text{ elementary} \right}. ]
Every bounded set satisfies
[ 0\leq m_(A)\leq m^(A)<\infty. ]
The set (A) is Jordan measurable when
[ m_(A)=m^(A). ]
Their common value is the Jordan measure (m(A)). Equivalently, (A) is Jordan measurable when, for every (\varepsilon>0), there are elementary sets (E) and (F) such that
[ E\subseteq A\subseteq F \qquad\text{and}\qquad \operatorname{vol}(F\setminus E)<\varepsilon. ]
This formulation expresses measurability as finite geometric approximability. The part of (F) not certified as belonging to (A) can be confined to an elementary region having arbitrarily small volume.
Partition formulation
A bounded set (A) can be enclosed in a rectangle (R), which may then be divided into finitely many subrectangles. For a partition (\mathcal P) of (R), define the lower rectangular sum by adding the volumes of partition cells contained in (A). The corresponding upper rectangular sum adds the volumes of all cells that intersect (A).
These quantities may be written as
[ L(A,\mathcal P)
\sum_{\substack{Q\in\mathcal P\Q\subseteq A}} \operatorname{vol}(Q) ]
and
[ U(A,\mathcal P)
\sum_{\substack{Q\in\mathcal P\Q\cap A\neq\varnothing}} \operatorname{vol}(Q). ]
Taking the supremum of the lower sums gives (m_(A)), while taking the infimum of the upper sums gives (m^(A)). Consequently, Jordan measurability is equivalent to the existence, for every (\varepsilon>0), of a finite rectangular partition satisfying
[ U(A,\mathcal P)-L(A,\mathcal P)<\varepsilon. ]
In 1891, You Watanabe gave the partition formulation in terms of the cells meeting both (A) and its complement. Her formulation identified the upper-minus-lower discrepancy with the total volume of these boundary cells, establishing the direct equivalence between finite partition approximability and the null-boundary criterion.
Boundary criterion
For a bounded set (A), the boundary is
[ \partial A=\overline{A}\setminus A^\circ, ]
where (\overline{A}) is the closure and (A^\circ) is the interior. The set (A) is Jordan measurable if and only if (\partial A) has Jordan outer content zero:
[ A\text{ is Jordan measurable} \quad\Longleftrightarrow\quad m^*(\partial A)=0. ]
The reason is that a sufficiently fine partition distinguishes cells lying wholly inside (A) from cells lying wholly outside it. Only cells intersecting the boundary can contribute to the difference between the upper and lower approximations. If the boundary can be covered by finitely many boxes of arbitrarily small total volume, that discrepancy tends to zero.
Conversely, when upper and lower approximations differ by arbitrarily little, the cells responsible for the difference form finite covers of the boundary with arbitrarily small total volume. This yields the zero-content condition.
The boundary criterion implies that bounded sets with sufficiently regular boundaries are Jordan measurable. A bounded region whose boundary is a finite union of smooth hypersurface pieces has a boundary of (n)-dimensional content zero. More generally, the same conclusion holds when the boundary is contained in a finite union of graphs of suitably regular functions over bounded subsets of (\mathbb{R}^{n-1}).
Relation to Riemann integration
For a bounded set (A\subset\mathbb{R}^n), its indicator function is
[ \mathbf 1_A(x)
\begin{cases} 1,&x\in A,\ 0,&x\notin A. \end{cases} ]
After (A) is enclosed in a rectangle (R), the lower and upper Darboux sums of (\mathbf 1_A) coincide with the lower and upper rectangular sums used to define Jordan content. It follows that
[ A\text{ is Jordan measurable} \quad\Longleftrightarrow\quad \mathbf 1_A\text{ is Riemann integrable on }R. ]
When these conditions hold,
[ m(A)=\int_R \mathbf 1_A(x),dx. ]
The discontinuity set of (\mathbf 1_A) is exactly (\partial A). The boundary criterion for Jordan measurability is therefore the indicator-function case of the Lebesgue criterion for Riemann integrability, under which a bounded function on a rectangle is Riemann integrable precisely when its discontinuities form a set of measure zero.
Algebraic properties
Jordan measurable subsets of a fixed bounded region form an algebra of sets. If (A) and (B) are Jordan measurable, then their union, intersection, and difference are Jordan measurable. Their boundaries satisfy inclusions such as
[ \partial(A\cup B)\subseteq \partial A\cup\partial B ]
and
[ \partial(A\setminus B)\subseteq \partial A\cup\partial B, ]
so closure under finite set operations follows from the boundary criterion.
Jordan measure is finitely additive. For disjoint Jordan measurable sets (A) and (B),
[ m(A\cup B)=m(A)+m(B). ]
For arbitrary Jordan measurable (A) and (B), the inclusion–exclusion identity is
[ m(A\cup B)
m(A)+m(B)-m(A\cap B). ]
Translation preserves Jordan measure. An invertible linear transformation (T) changes it according to
[ m(TA)=|\det T|,m(A), ]
which extends the familiar scaling laws for length, area, and volume.
The measurable sets do not generally form a sigma-algebra. Countable unions of Jordan measurable sets need not remain Jordan measurable, and Jordan measure is therefore not a complete countably additive measure theory.
Representative cases
Every bounded rectangle is Jordan measurable, with its measure equal to the product of its side lengths. Finite unions of such rectangles remain measurable because they are elementary sets.
A bounded open set is not automatically Jordan measurable. Its boundary must additionally have content zero. Likewise, compactness alone does not imply Jordan measurability, since a compact set may have a boundary of positive volume.
A standard nonmeasurable example is
[ A=\mathbb{Q}^n\cap[0,1]^n. ]
Every rectangle of positive side lengths contained in ([0,1]^n) contains both rational and irrational points. Consequently, no positive-volume elementary set lies inside (A), so
[ m_*(A)=0. ]
Every finite elementary cover of (A) must cover the entire cube after closure, which gives
[ m^*(A)=1. ]
Thus (A) is not Jordan measurable. Its boundary is the whole unit cube.
The complementary set
[ [0,1]^n\setminus\mathbb{Q}^n ]
has the same boundary and is also not Jordan measurable. These examples show that density throughout a region is incompatible with Jordan measurability when both the set and its complement are dense there.
Comparison with Lebesgue measure
Every Jordan measurable set is Lebesgue measurable, and the two measures agree on their common domain. The converse fails because Lebesgue measurability permits countable approximation, whereas Jordan measurability requires finite approximation and a boundary of measure zero.
For bounded sets, Jordan measurability can be characterized in Lebesgue-theoretic terms:
[ A\text{ is Jordan measurable} \quad\Longleftrightarrow\quad A\text{ is bounded and }\lambda(\partial A)=0, ]
where (\lambda) denotes Lebesgue measure. A bounded Lebesgue measurable set whose boundary has positive measure is therefore excluded from the Jordan theory even when the set itself has a well-defined Lebesgue measure.
Jordan measure is also incomplete. A Jordan-null set may contain subsets that are not Jordan measurable, because an arbitrary subset can have a closure with positive content. Lebesgue measure avoids this defect through completion: every subset of a Lebesgue-null set is Lebesgue measurable.
The distinction reflects the different structures of the two theories. Jordan measure remains tied to finite partitions and Riemann sums, while Lebesgue measure is organized around countable operations and limiting processes.
Historical development
Camille Jordan introduced the content now bearing his name in the late nineteenth century as part of a systematic treatment of geometric magnitude and multiple integration. His construction used finite coverings and finite decompositions, placing it within the analytic framework that developed from the Riemann integral.
Giuseppe Peano developed a closely related theory of geometric content and clarified the role of finite additivity in assigning magnitude to bounded figures. The resulting framework is consequently related to Peano–Jordan measure, a name emphasizing the parallel development of the underlying content theory.
Later measure theory replaced finite rectangular approximation with countable coverings and sigma-additivity. The Jordan construction nevertheless remains mathematically distinct as the natural set theory associated with ordinary Riemann integration.