# The Unapologetic Mathematician

## Jordan Content and Boundaries

As a first exercise working with Jordan content, let’s consider how it behaves at the boundary of a region.

I’ve used this term a few times when it’s been pretty clear from context, but let me be clear. We know about the interior and closure of a set, and particularly of a subset of $\mathbb{R}^n$. The boundary of such a set will consist of all the points in the closure of the set, but not in its interior. That is, we have $\mathrm{Cl}(S)=\mathrm{int}(S)\uplus\partial S$

That is, a point $x$ is in $\partial S$ if $x\in\mathrm{Cl}(S)$, but any neighborhood of $x$ contains points not in $S$.

So let’s put $S$ inside a box $[a,b]$ and partition the box with a partition $P$. When we calculate $\overline{J}(P,S)$, we include all the subintervals that we do when we calculate $\underline{J}(P,S)$, along with some other subintervals which contain both points within $\mathrm{int}(S)$ and points not in $\mathrm{int}(S)$. I say that each of these are exactly the subintervals which contain points in $\partial S$. Indeed, if a subinterval contains a point of $\partial S$ it cannot be included when calculating $\underline{J}(P,S)$, but must be included when calculating $\overline{J}(P,S)$. Inversely, if a subinterval contains no point of $\partial S$ then it is either contained completely within $\mathrm{int}(S)$ — and is included in both calculations — or it is contained completely within the complement of $\mathrm{Cl}(S)$ — and is contained in neither computation. Thus we have the relation $\displaystyle\overline{J}(P,S)=\underline{J}(P,S)+\overline{J}(P,\partial S)$

which we rewrite $\displaystyle\overline{J}(P,\partial S)=\overline{J}(P,S)-\underline{J}(P,S)$

We can then pass to infima to find $\displaystyle\overline{c}(\partial S)\geq\overline{c}(S)-\underline{c}(S)$

Check this carefully to see how the equality is weakened to an inequality.

On the other hand, given any $\epsilon>0$ we can pick partitions $P_1$ so that $\overline{J}(P_1,S)<\overline{c}(S)+\frac{\epsilon}{2}$ and $P_2$ so that $\underline{J}(P_2,S)>\underline{c}(S)-\frac{\epsilon}{2}$. We let $P$ be a common refinement of $P_1$ and $P_2$, which will then satisfy both of these inequalities. We find $\displaystyle\overline{c}(S)\leq\overline{J}(P,\partial S)=\overline{J}(P,S)-\underline{J}(P,S)<\overline{c}(S)-\underline{c}(S)+\epsilon$

since $\epsilon$ is arbitrary, we find $\overline{c}(\partial S)\leq\overline{c}(S)-\underline{c}(S)$.

Together with the previous result, we conclude that $\overline{c}(\partial S)=\overline{c}(S)-\underline{c}(S)$. In particular, we find that $S$ is Jordan measurable if and only if $c(\partial S)=\overline{c}(\partial S)=0$.

December 4, 2009 Posted by | Analysis, Measure Theory | 6 Comments