Peano–Jordan measure

In mathematics, the Peano measure (also known as the Peano 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 Peano content it should be well-behaved in a restrictive sense. For this reason, it is now more common to work with the Lebesgue measure, which is an extension of the Peano content to a larger class of sets. Historically speaking, Peano content came first, towards the end of the nineteenth century. For historical reasons, the term Peano measure is now well-established for this set function, despite the fact that it is not a true measure in its modern definition, since sets with Peano content do not form a σ-algebra. For example, singleton sets $\{x\}_{x\in \mathbb {R} }$in $\mathbb {R}$ each have a Peano content of 0, while $\mathbb {Q} \cap [0,1]$, a countable union of them, does not have Peano content. For this reason, some authors use the term Peano content.

The Peano measure is named after its originators, the Italian mathematician Giuseppe Peano.

Peano content of "simple sets"

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.

Consider Euclidean space $\mathbb {R} ^{n}.$ Peano content is first defined on Cartesian products of bounded half-open intervals

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

that are closed at the left and open at the right with all endpoints $a_{i}$ and $b_{i}$ finite real numbers (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. The Peano content of such a rectangle is defined 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 unions of rectangles,

$$S=C_{1}\cup C_{2}\cup \cdots \cup C_{k}$$

for any $k\geq 1.$

One cannot define the Peano content 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, and then one defines the Peano content $m(S)$ as the sum of measures of the disjoint rectangles.

One can show that this definition of the Peano content 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

A set (represented in the picture by the region inside the blue curve) has Peano content 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).

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 sets with Peano content is still very limited. The key step is then defining a bounded set to have Peano content 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 Peano content as

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

and its outer Peano content as

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

where the infimum and supremum are taken over simple sets $S.$ The set $B$ is said to be a set with Peano content if the inner measure of $B$ equals the outer measure. The common value of the two measures is then simply called the Peano content of $B$. The Peano content is the set function that sends sets with Peano content to the value of that content.

It turns out that all rectangles (open or closed), as well as all balls, simplexes, etc., have Peano content. Also, if one considers two continuous functions, the set of points between the graphs of those functions has Peano content as long as that set is bounded and the common domain of the two functions has Peano content. Any finite union and intersection of sets with Peano content has Peano content, as well as the set difference of any two sets with Peano content. A compact set does not necessarily have Peano content. For example, the ε-Cantor set is not. Its inner Peano content vanishes, since its complement is dense; however, its outer Peano content does not vanish, since it cannot be less than (in fact, is equal to) its Lebesgue measure. Also, a bounded open set does not necessarily have Peano content. A bounded set has Peano content if and only if its indicator function is Riemann-integrable, and the value of the integral is its Peano content.[1]

Equivalently, for a bounded set $B$ the inner Peano content of $B$ is the Lebesgue measure of the topological interior of $B$ and the outer Peano content is the Lebesgue measure of the closure. From this it follows that a bounded set has Peano content if and only if its topological boundary has Lebesgue measure zero. (Or equivalently, if the boundary has Peano content zero; the equivalence holds due to compactness of the boundary.)

References

• Emmanuele DiBenedetto (2002). Real analysis. Basel, Switzerland: Birkhäuser. ISBN 0-8176-4231-5.
• Richard Courant; Fritz John (1999). Introduction to Calculus and Analysis Volume II/1: Chapters 1–4 (Classics in Mathematics). Berlin: Springer. ISBN 3-540-66569-2.

External links

• Derwent, John. "Jordan Measure". MathWorld.
• Terekhin, A.P. (2001) [1994], "Jordan measure", Encyclopedia of Mathematics, EMS Press