Jordan Content is Monotone

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.

$\blacksquare$