View source for Jordan measure

''of a parallelepiped

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in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054350/j0543502.png" />''

The volume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054350/j0543503.png" /> of this parallelepiped. The following are defined for a bounded set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054350/j0543504.png" />: the outer Jordan measure

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and the inner Jordan measure

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where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054350/j0543507.png" /> are pairwise disjoint (here the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054350/j0543508.png" /> are parallelepipeds of the form ). A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054350/j0543509.png" /> is said to be Jordan measurable (squarable for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054350/j05435010.png" />, cubable for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054350/j05435011.png" />) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054350/j05435012.png" /> or, equivalently, if

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054350/j05435014.png" />. In this case, the Jordan measure is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054350/j05435015.png" />. A bounded set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054350/j05435016.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054350/j05435017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054350/j05435018.png" /> (the closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054350/j05435019.png" />, cf. [[Closure of a set|Closure of a set]]) and is equal to the [[Borel measure|Borel measure]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054350/j05435020.png" />. 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.