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''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|Lebesgue measure]] zero).

The concept of this measure was introduced by G. Peano [[#References|[1]]] and C. Jordan [[#References|[2]]]. The outer measure is the same for  $  E $
and  $  \overline{E}\; $(
the closure of  $  E $, 
cf. [[Closure of a set|Closure of a set]]) and is equal to the [[Borel measure|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|Squarability]].

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

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