Jordan Regions I
We need to think much harder than we did in 1D about the kinds of regions we can sensibly integrate over in higher dimensions.
Given any set \(E\) contained in a rectangle \(R\), if \(\mathcal P\) is a partition of \(R\), we define
\[ V(E,{\mathcal P}) := \mathop{\sum_{R' \in {\mathcal P}}}_{R' \cap E \neq \emptyset} |R'|. \]
Proposition (Basic Inequalities)
- If \(E \subset F \subset R\) and \({\mathcal P}\) is a partition of \(R\), then \(V(E,{\mathcal P}) \leq V(F,{\mathcal P})\).
- If \(E,F \subset R\) and \({\mathcal P}\) is a partition of \(R\), then\[{}V(E \cap F,{\mathcal P}){}\]\[{}\leq \min\{ V(E,{\mathcal P}), V(F,{\mathcal P})\}{}\]and\[{}V(E \cup F,{\mathcal P}){}\]\[{}\leq V(E,{\mathcal P}) + V(F,{\mathcal P}).{}\]
- If \({\mathcal P}'\) is a refinement of \({\mathcal P}\), then \(V(E,{\mathcal P}') \leq V(E,{\mathcal P})\).
We say a set \(E \subset {\mathbb R}^n\) has Jordan content zero if and for every \(\epsilon > 0\), there is a box \(R\) containing \(E\) and a partition \({\mathcal P}\) of \(R\) such that \(V(E,{\mathcal P}) < \epsilon\). (Wade calls these “volume zero” sets.) Note that the containing box \(R\) can always be enlarged if desired by first “padding” the partition with very thin boxes around the boundary of \(R\). We need to do this step because \(E\) may have points in common with the boundary, so we need any additional boxes which touch the boundary of \(R\) to have small volume.
Lemma
A bounded set \(E \subset R\) has Jordan content zero if and only if for every \(\epsilon > 0\), there exists a finite set of closed cubes \(Q_1,\ldots,Q_N\) of equal size such that \(E \subset Q_1 \cup \cdots \cup Q_N\) and \(|Q_1| + \cdots + |Q_N| < \epsilon\).
Proof