polyrectangle

A polyrectangle $P$ in ${ℝ}^n$ is a finite collection $P=\{R_1,\mathrm{\dots },R_N\}$ of compact rectangles $R_i\subset {ℝ}^n$ with disjoint interior.A compact rectangle $R_i$ is a Cartesian product of compact intervals: $R_i=[a_1^i,b_1^i]\times \mathrm{\cdots }\times [a_n^i,b_n^i]$ where (these are also called $n$-dimensional intervals).

The union of the compact rectangles of a polyrectangle $P$ is denoted by

$$\cup P:=\bigcup _{R\in P}R=R_1\cup \mathrm{\cdots }\cup R_N.$$

It is a compact subset of ${ℝ}^n$.

We can define the ($n$-dimensional) measure of $\cup P$ in a way.If $R=[a_1,b_1]\times \mathrm{\cdots }\times [a_n,b_n]$ is a rectangle we define the measure of $R$ as

$$\mathrm{meas}(R):=(b_1-a_1)\mathrm{\cdots }(b_n-a_n)$$

and define the measure of the polyrectangle $P$ as:

$$\mathrm{meas}(P):=\sum _{R\in P}\mathrm{meas}(R).$$

Moreover if we are given a bounded function $f:\cup P\to ℝ$ we can define the upper and lower Riemann sums of $f$ over $\cup P$ by

$$S^{*}(f,P):=\sum _{R\in P}\mathrm{meas}(R)\underset{x\in R}{sup}f(x),S_{*}(f,P):=\sum _{R\in P}\mathrm{meas}(R)\underset{x\in R}{inf}f(x).$$

Polyrectangles are then used to define the Peano Jordan measure of subsets of ${ℝ}^n$ and to define Riemann multiple integrals.To achieve this, it is useful to introduce the so called refinements. The family of rectangles $R_i$ which appear in the definition LABEL:defpoly are called a partition of $\overline{\cup P}$ in rectangles.It is clear that the set $\cup P$ can be represented by different polyrectangles. For example any rectangle $R$ can be split in $2^n$ smaller rectangles by dividing in two parts each of the $n$ intervals defining $R$.We claim that given two polyrectangles $P$ and $Q$ there exists a polyrectangle $S$ such that $(\cup P)\cup (\cup Q)\subset \cup S$ and such that given any rectangle $R$ in $P$ or $Q$, $R$ is the union of rectangles in $S$.