Inner Jordan Content is Monotone

Theorem

Let $A, B \subseteq \R^n$ be bounded subspaces of Euclidean $n$-space.

Suppose that $A \subseteq B$.

Then:

$\map {m_*} A \le \map {m_*} B$

where $m_*$ denotes the inner Jordan content.

Proof

Let $R \subseteq \R^n$ be a closed $n$-rectangle that contains $B$.

Then, by Subset Relation is Transitive, $R$ contains $A$ as well.

By Set Difference with Subset is Superset of Set Difference:

$R \setminus B \subseteq R \setminus A$

So, by Outer Jordan Content is Monotone:

$\map {m^*} {R \setminus B} \le \map {m^*} {R \setminus A}$

Therefore:

$\map V R - \map {m^*} {R \setminus B} \ge \map V R - \map {m^*} {R \setminus A}$

Hence the result, by definition of inner Jordan content.

$\blacksquare$