Peano–Jordan measure

In mathematics, the Jordan measure is an extension of the notion of size (length, area, volume) to shapes more complicated than, for example, a triangle, disk, or parallelipiped.

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, Jordan measure came first, towards the end of the nineteenth century.

Jordan measure of "simple sets"

Consider the Euclidean space R^{n. One starts by considering products of bounded intervals}

$C=[a_{1},b_{1})\times [a_{2},b_{2})\times \cdots \times [a_{n},b_{n})$

which are closed at the left end and open at the right end. 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, which are a finite 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 rectangles, rectangles which this time are mutually 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. 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 if it is well-approximated by piecewise-constant functions.

Formally, for a bounded set B, define its inner Jordan measure as

$m_{*}(B)=\sup _{S\subset B}m(S)$

and its outer measure as

$m^{*}(B)=\inf _{S\supset B}m(S)$

where the infimum and supremum are taken over simple sets S. One says that B is Jordan measurable if the inner measure of B equals the outer measure, and then the Jordan measure of B is the common value of these two measures.

It turns out that all rectangles (with or without boundary), as well 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. Any finite union and intersection of Jordan sets is measurable, as well as the set difference of any two Jordan measurable sets. One can also prove that a bounded set is Jordan measurable if and only if its boundary is Jordan measurable and has Jordan measure zero.

The Lebesgue measure

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. Intuitively however, the set of rational numbers is a "small" set, as it is countable, and it should have "size" zero. That is indeed true, but 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.

External links

• Jordan Measure—MathWorld