Category:Inner Jordan Content
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This category contains results about Inner Jordan Content.
Definitions specific to this category can be found in Definitions/Inner Jordan Content.
Let $M \subseteq \R^n$ be a bounded subspace of Euclidean $n$-space.
Let:
- $\ds R = \prod_{i \mathop = 1}^n \closedint {a_i} {b_i}$
be a closed $n$-rectangle that contains $M$.
Let $V_R \in \R_{\ge 0}$ be defined as:
- $\ds V_R = \prod_{i \mathop = 1}^n \paren {b_i - a_i}$
The inner Jordan content of $M$ is defined and denoted as:
- $\map {m_*} M = V_R - \map {m^*} {R \setminus M}$
where:
- $\map {m^*} {R \setminus M}$ denotes the outer Jordan content of $R \setminus M$
That is, the inner Jordan content of $M$ is defined to be the outer Jordan content of the complement of $M$, relative to some fixed $R \supseteq M$.
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