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| Title of object: |
polyrectangle |
| Canonical Name: |
Polyrectangle |
| Type: |
Definition |
| Created on: |
2005-02-18 08:48:02 |
| Modified on: |
2005-03-09 10:12:19 |
| Classification: |
msc:26A42 |
| Defines: |
Riemann sums on polyrectangles |
Preamble:
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Content:
A polyrectangle $P$ in $\R^n$ is a finite collection $P=\{R_1,\ldots,R_N\}$ of compact rectangles $R_i\subset \R^n$ with disjoint interior.
A compact rectangle $R_i$ is a cartesian product of compact intervals: $R_i=[a_1^i,b_1^i]\times \ldots \times [a_n^i,b_n^i]$ where $a_j^i<b_j^i$.
The union of the rectangles of a polyrectangle $P$ is denoted by
\[
\cup P := \bigcup_{R\in P} R = R_1 \cup \ldots \cup R_N
\]
and is a compact subset of $\R^n$.
We can define the ($n$-dimensional) measure of $\cup P$ in a simple way.
If $R=[a_1,b_1]\times \ldots \times [a_n,b_n]$ is a rectangle we define the measure of $R$ as
\[
\mathrm{meas}(R) := (b_1-a_1)\cdots (b_n-a_n)
\]
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 $f\colon \cup P\to\mathbb R$ we can define the \emph{upper} and \emph{lower Riemann sums} of $f$ over $\cup P$ by
\[
S^*(f,P) := \sum_{R\in P} \sup_{x\in R_i},\qquad
S_*(f,P) := \sum_{R\in P} \inf_{x\in R_i}.
\]
Polyrectangles are then used to define the Peano Jordan measure of subsets of $\mathbb R^n$ and to define Riemann multiple integrals.
To achieve this, it is useful to introduce the so called \emph{refinements}. The family of rectangles $R_i$ which appear in the definition~\ref{defpoly} are called a \emph{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 every 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 $S$. |
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