Jordan content

The volume assigned to a finite union of rectangles, used as a precursor to measure.

Jordan content

A (closed) rectangle in $\mathbb{R}^k$ is a set of the form

R=\prod_{j=1}^k [a_j,b_j],\qquad a_j\le b_j,

with volume

\operatorname{vol}(R)=\prod_{j=1}^k (b_j-a_j).

An elementary set is a finite union of rectangles. If an elementary set $E$ is written as a finite union of pairwise disjoint rectangles $R_1,\dots,R_N$ , its Jordan content (also called content) is

c(E):=\sum_{i=1}^N \operatorname{vol}(R_i).

(One checks that for elementary sets this is well-defined: different disjoint-rectangle decompositions yield the same total volume.)

Jordan content is a finite-additive “volume” defined on elementary sets, and it provides a convenient language for coverings in measure-zero arguments without requiring full measure theory.

Examples:

For a rectangle $R$ , $c(R)=\operatorname{vol}(R)$ .
In $\mathbb{R}$ , if $E=[0,1]\cup[2,3]$ (disjoint union), then $c(E)=1+1=2$ .
If two rectangles overlap, one first decomposes their union into disjoint rectangles to compute content additively.