Well-Defined Jordan Content Equals Content

This article needs to be linked to other articles. In particular: category You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{MissingLinks}} from the code.

Theorem

Let $M$ be a bounded subspace of Euclidean space.

Let the Jordan content of $M$ be $\map m M$.

Then the content $\map V M = \map m M$.

Proof

Let $C$ be a finite covering of $M$.

By Common Notion $5$, $\map V C \ge \map V M$.

Therefore, $\map V M$ is a lower bound of all $\map V C$.

So by the definition of greatest lower bound:

$\map V M \le \map {m^*} M$

Let $D$ be a finite covering of $S \setminus M$.

By the same reasoning:

$\map V {S \setminus M} \le \map {m^*} {S \setminus M}$

But:

Therefore:

$\map V S - \map {m^*} {S \setminus M} \le \map V M \le \map {m^*} M$

But by hypothesis:

$\map V S - \map {m^*} {S \setminus M} = \map {m^*} M = \map m M$

So it follows that:

$\map V M = \map m M$

$\blacksquare$