# 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  and C. Jordan . 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.

How to Cite This Entry:
Jordan measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jordan_measure&oldid=47470
This article was adapted from an original article by A.P. Terekhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article