Peano–Jordan measure

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.

[1] For this reason, some authors[2] 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 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

finite real numbers (half-open intervals is a technical choice; as we see below, one can use closed or open intervals if preferred).

The Jordan measure of such a rectangle is defined to be the product of the lengths of the intervals:

Next, one considers simple sets, sometimes called polyrectangles, which are finite unions of rectangles,

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.

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.

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.

Notice that a set which is a product of closed intervals,

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 if it is well-approximated by piecewise-constant functions.

where the infimum and supremum are taken over simple sets

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.

A compact set is not necessarily Jordan measurable.

Also, a bounded open set 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 measure.

is the Lebesgue measure of the topological interior of

This last property greatly limits the types of sets which are Jordan measurable.

For example, the set of rational numbers contained in the interval [0,1] is then not Jordan measurable, as its boundary is [0,1] which is not of Jordan measure zero.

A simple set is, by definition, a union of (possibly overlapping) rectangles.
The simple set from above decomposed as a union of non-overlapping rectangles.
A set (represented in the picture by the region inside the blue curve) is Jordan measurable if and only if it can be well-approximated both from the inside and outside by simple sets (their boundaries are shown in dark green and dark pink respectively).