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 in each have a Peano content of 0, while , a countable union of them, does not have Peano content.[1] For this reason, some authors[2] use the term Peano
content.
The Peano measure is named after its originators, the Italian mathematician
Giuseppe Peano.[3]
that are closed at the left and open at the right with all endpoints and 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 -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:
Next, one considers simple sets, sometimes called polyrectangles, which are finite
unions of rectangles,
for any
One cannot define the Peano content of 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.
Luckily, any such simple set 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 as the sum of measures of the disjoint rectangles.
One can show that this definition of the Peano content of is independent of the representation of 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,
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 define its inner Peano content as
and its outer Peano content as
where the
infimum and
supremum are taken over simple sets The set is said to be a set with Peano content if the inner measure of equals the outer measure. The common value of the two measures is then simply called the Peano content of . 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 the inner Peano content of is the Lebesgue measure of the
topological interior of and the outer Peano content is the Lebesgue measure of the
closure.[4] 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
^While a set whose measure is defined is termed measurable, there is no commonly accepted term to describe a set whose Peano content is defined. Munkres (1991) suggests the term "rectifiable" as a generalization of the use of this term to describe curves. Other authors have used terms including "admissible" (Lang, Zorich); "pavable" (Hubbard); "have content" (Burkill); "contented" (Loomis and Sternberg).
^Munkres, J. R. (1991). Analysis on Manifolds. Boulder, CO: Westview Press. p. 113.
ISBN0-201-31596-3.
Emmanuele DiBenedetto (2002). Real analysis. Basel, Switzerland: Birkhäuser.
ISBN0-8176-4231-5.
Richard Courant; Fritz John (1999). Introduction to Calculus and Analysis Volume II/1: Chapters 1–4 (Classics in Mathematics). Berlin: Springer.
ISBN3-540-66569-2.