Jordan measure

of a parallelepiped

$$ \tag{* } \Delta = \{ {x \in \mathbf R ^ {n} } : { a _ {i} \leq x _ {i} \leq b _ {i} ,\ a _ {i} < b _ {i} , i = 1 \dots n } \} $$

in $ \mathbf R ^ {n} $

The volume $ m \Delta = \prod _ {i=} 1 ^ {n} ( b _ {i} - a _ {i} ) $ of this parallelepiped. The following are defined for a bounded set $ E \subset \mathbf R ^ {n} $: the outer Jordan measure

$$ m _ {e} E = \inf \sum _ { j= } 1 ^ { k } m \Delta _ {j} ,\ \ \cup _ { j= } 1 ^ { k } \Delta _ {j} \supset E ,\ \ k = 1 , 2 \dots $$

and the inner Jordan measure

$$ m _ {i} E = \sup \sum _ { j= } 1 ^ { k } m \Delta _ {j} ,\ \ E \supset \Delta _ {j} , $$

where the $ \Delta _ {j} $ are pairwise disjoint (here the $ \Delta _ {j} $ are parallelepipeds of the form ). A set $ E $ is said to be Jordan measurable (squarable for $ n = 2 $, cubable for $ n \geq 3 $) if $ m _ {e} E = m _ {i} E $ or, equivalently, if

$$ m _ {e} E + m _ {e} ( \Delta \setminus E ) = m \Delta , $$

where $ \Delta \supset E $. In this case, the Jordan measure is $ m E = m _ {e} E = m _ {i} E $. A bounded set $ E \subset \mathbf R ^ {n} $ is Jordan measurable if and only if its boundary has Jordan measure zero (or, equivalently, if its boundary has Lebesgue measure zero).

The concept of this measure was introduced by G. Peano [1] and C. Jordan [2]. The outer measure is the same for $ E $ and $ \overline{E}\; $( the closure of $ E $, cf. Closure of a set) and is equal to the Borel measure of $ \overline{E}\; $. The Jordan-measurable sets form a ring of sets on which the Jordan measure is a finitely-additive function. See also Squarability.

References

[1]	G. Peano, "Applicazioni geometriche del calcolo infinitesimale" , Bocca (1887)
[2]	C. Jordan, "Remarques sur les intégrales définies" J. Math. Pures Appl. , 8 (1892) pp. 69–99
[3]	S.M. Nikol'skii, "A course of mathematical analysis" , 2 , MIR (1977) (Translated from Russian)
[4]	A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian)

Comments

Jordan measure (also called Jordan content) is now mainly of historical interest, since it is simply the restriction of Lebesgue measure to the ring of bounded Lebesgue-measurable sets having boundary of measure 0.