Outer Jordan Content is Subadditive

Theorem

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

Then:

$\map {m^*} {A \cup B} \le \map {m^*} A + \map {m^*} B$

where $m^*$ denotes the outer Jordan content.

Proof

Let $\epsilon > 0$ be arbitrary.

By Characterizing Property of Infimum of Subset of Real Numbers, select:

$C$ to be a finite covering of $A$ by closed $n$-rectangles such that:

$\ds \sum_{R \mathop \in C} \map V R < \map {m^*} A + \frac \epsilon 2$

$D$ to be a finite covering of $B$ by closed $n$-rectangles such that:

$\ds \sum_{R \mathop \in D} \map V R < \map {m^*} B + \frac \epsilon 2$

Define $E = C \cup D$.

By Union of Finite Sets is Finite, $E$ is a finite set of closed $n$-rectangles.

So, by Union of Covers is Cover of Union:

$C \cup D$ is a finite covering of $A \cup B$ by closed $n$-rectangles.

We have:

\(\ds \map {m^*} {A \cup B}\)

\(\le\)

\(\ds \sum_{R \mathop \in E} \map V R\)

Definition of Outer Jordan Content

\(\ds \)

\(=\)

\(\ds \sum_{R \mathop \in C \cup D} \map V R\)

\(\ds \)

\(\le\)

\(\ds \sum_{R \mathop \in C} \map V R + \sum_{R \mathop \in D} \map V R\)

$\map V R$ is non-negative, so any double-counted elements only increase the sum

\(\ds \)

\(<\)

\(\ds \map {m^*} A + \map {m^*} B + \epsilon\)

As $\epsilon > 0$ was arbitrary, it follows from Real Plus Epsilon that:

$\map {m^*} {A \cup B} \le \map {m^*} A + \map {m^*} B$

$\blacksquare$