Integration Partitions and Grids
1. Definitions
Let sets in \({\mathbb R}^n\) of the form \(R = [a_1,b_1] \times \cdots \times [a_n , b_n]\) be called boxes (or simply rectangles). The volume of \(R\) is defined to equal the product of its side lengths, i.e.,
\[ |R| := \prod_{j=1}^n (b_j - a_j). \]
Two boxes \(R_1\) and \(R_2\) are called nonoverlapping when \(R_1 \cap R_2\) has no interior.
We say \(\mathcal P\) is a partition of box \(R \subset {\mathbb R}^n\) when \(\mathcal P\) is a finite set of pairwise nonoverlapping boxes contained in \(R\) whose union is \(R\).
A grid is a special partition \({\mathcal P}\) of \(R\) obtained by partitioning each interval \([a_1,b_1],\ldots,[a_n,b_n]\) by partitions \({\mathcal P}_1,\ldots,{\mathcal P}_n\), respectively and letting \(P\) consist of all possible products \(I_1 \times \cdots \times I_n\) where \(I_j \in {\mathcal P}_j\) for each \(j=1,\ldots,n\).
2. Refinements
Given two partitions \(\mathcal P\) and \(\mathcal P'\) of the same box \(R\), we say that \(\mathcal P'\) refines \(\mathcal P\) when every box \(R''\) in \({\mathcal P}'\) is entirely contained in some box \(R' \in {\mathcal P}\).
Theorem (Adapted Refinements)
Given any finite collection of boxes \(R_1,\ldots,R_N \subset {\mathbb R}^n\) contained in some box \(R\), There is a grid partition \(\mathcal G\) of \(R\) such that for each \(i \in \{1,\ldots,N\}\), the elements of \(\mathcal G\) which intersect the interior of \(R_i\) form a grid partition \({\mathcal G}_i\) of \(R_i\), i.e., they are nonoverlapping and have union equal exactly to \(R_i\).
Proof
Corollary (Common Grid Refinements of Partitions)
Given any finite (positive) number of partitions \({\mathcal P}_1,\ldots,{\mathcal P}_N\), of \(R\), there is a grid partition \(\mathcal G\) of \(R\) which is a refinement of \({\mathcal P}_j\) for each \(j\). (As in the one-dimensional theory, we say that \(\mathcal G\) is a common refinement of \({\mathcal P}_1,\ldots,{\mathcal P}_N\).) Moreover, if \(R' \in {\mathcal P}_j\), then the elements of \(\mathcal G\) which overlap with \(R'\) form a grid partition of \(R'\).
Proof
3. Volume Equalities and Inequalities
Theorem (On Volumes)
- If \(\mathcal P\) is a partition of the box \(R\), then\[ |R| = \sum_{R' \in \mathcal P} |R'|. \]
- If \(R_1,\ldots R_N\) are any boxes in \({\mathbb R}^n\) and if \(\mathcal G\) is a grid partition adapted to the boxes \(R_1,\ldots,R_N\) as above, then\[ \mathop{\sum_{R' \in \mathcal G}}_{R' \subset \bigcup_j R_j} |R'| \leq \sum_{j=1}^N |R_j| \]with equality if and only if the \(R_j\)'s are nonoverlapping.
Proof
4. Approximate Cubes
We will call a set \(R := [a_1,b_1] \times \cdots \times [a_n,b_n]\) an approximate cube of side length \(\delta > 0\) when \(\delta \leq b_k - a_k \leq 2 \delta\) for each \(k \in \{1,\ldots,n\}\).
Theorem (Refining with Approximate Cubes)
Given any grid partition \(\mathcal G\) of a rectangle \(R\), it has a refinement \({\mathcal G}'\), also a grid partition such that there is some \(\delta > 0\) for which every \(R' \in {\mathcal G}'\) is an approximate cube of side length \(\delta\).
Proof
Exercise
Suppose that \(R_0\) is any cube and \(\mathcal P\) is any partition of it. Show that there is a constant \(C\) depending only on \(R_0\) and \(\mathcal P\) such that every grid partition \(\mathcal G\) of \(R_0\) admits some partition \(\mathcal{P}'\) of \(R_0\) which is a common refinement of \(\mathcal P\) and \(\mathcal G\) such that
\[ \sum_{R \in \mathcal G \setminus \mathcal{P}'} |R| = \sum_{R' \in \mathcal{P}' \setminus \mathcal G} |R'| \leq C || \mathcal{G}|| \]
where \(||\mathcal{G}||\) is the diameter of the largest box in \(\mathcal{G}\).
Hint
Use the common grid refinement of \(\mathcal{P}\) and \(\mathcal{G}\) and prove a bound on the sum of the volumes of the boxes of \(\mathcal{G}\) which get subdivided in this procedure. This is analogous to an argument appearing in the proof of equivalence of Riemann and Darboux integrals.