Jordan Content is Monotone
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Theorem
Let $A, B \subseteq \R^n$ be bounded subspaces of Euclidean $n$-space.
Suppose that $A \subseteq B$.
Further suppose that the Jordan content of both $A$ and $B$ exists.
Then:
- $\map m A \le \map m B$
where $m$ denotes the Jordan content.
Proof
By definition of the Jordan content:
- $\map m A = \map {m^*} A$
- $\map m B = \map {m^*} B$
where $m^*$ denotes the outer Jordan content.
The result follows from Outer Jordan Content is Monotone.
$\blacksquare$