Definition:Outer Jordan Content
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Definition
Let $M \subseteq \R^n$ be a bounded subspace of Euclidean $n$-space.
The outer Jordan content of $M$ is defined as denoted as:
- $\ds \map {m^*} M = \inf_C \sum_{R \mathop \in C} \map V R$
where:
- the infimum $\ds \inf_C$ is taken over all finite coverings $C$ of $M$ by closed $n$-rectangles
 | This article, or a section of it, needs explaining. In particular: The ordered set and ordering relation need to be clarified. Presumably this is the "set of all finite coverings of $M$" and the subset relation. Does it need to be proved that the "set of all finite coverings of $M$" actually admits an infimum? You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code. |
- $R = \closedrect {\mathbf a} {\mathbf b}$ denotes a closed $n$-rectangle of $C$ for $\mathbf a, \mathbf b \in \R^n$
- $\map V R$ is defined as being:
- $\map V R := \ds \prod_{i \mathop = 1}^n \paren {b_i - a_i}$
- where $a_i$, $b_i$ are the coordinates of $\mathbf a = \tuple {a_1, a_2, \dotsc, a_n}$ and $\mathbf b = \tuple {b_1, b_2, \dotsc, b_n}$.
| This article, or a section of it, needs explaining. In particular: Does the same apply to $R$ in this context as it does in Definition:Inner Jordan Content? It is still unclear quite how all this fits together. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code. |
Also known as
The outer Jordan content is often referred to as the outer Jordan measure, but this is a misnomer as it does not constitute an outer measure.
Also see
- Results about outer Jordan content can be found here.
Source of Name
This entry was named for Marie Ennemond Camille Jordan.
Sources
- 1994: A. Shenitzer and J. Steprans: The Evolution of Integration (Amer. Math. Monthly Vol. 101, no. 1: pp. 66 – 72) www.jstor.org/stable/2325128
- Derwent, John. "Jordan Measure." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/JordanMeasure.html