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Version 15 |
| 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 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. |
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A \emph{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 $a_j^i<b_j^i$ (these are also called \emph{$n$-dimensional intervals}).
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A \emph{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 $a_j^i<b_j^i$.
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| The union of the rectangles of a polyrectangle $P$ is denoted by |
The union of the rectangles of a polyrectangle $P$ is denoted by |
| \[ |
\[ |
| \cup P := \bigcup_{R\in P} R = R_1 \cup \ldots \cup R_N. |
\cup P := \bigcup_{R\in P} R = R_1 \cup \ldots \cup R_N. |
| \] |
\] |
| It is a compact subset of $\R^n$. |
It is a compact subset of $\R^n$. |
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| We can define the ($n$-dimensional) measure of $\cup P$ in a \PMlinkescapetext{simple} way. |
We can define the ($n$-dimensional) measure of $\cup P$ in a \PMlinkescapetext{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 |
If $R=[a_1,b_1]\times \ldots \times [a_n,b_n]$ is a rectangle we define the measure of $R$ as |
| \[ |
\[ |
| \mathrm{meas}(R) := (b_1-a_1)\cdots (b_n-a_n) |
\mathrm{meas}(R) := (b_1-a_1)\cdots (b_n-a_n) |
| \] |
\] |
| and define the measure of the polyrectangle $P$ as: |
and define the measure of the polyrectangle $P$ as: |
| \[ |
\[ |
| \mathrm{meas}(P) := \sum_{R\in P} \mathrm{meas}(R). |
\mathrm{meas}(P) := \sum_{R\in P} \mathrm{meas}(R). |
| \] |
\] |
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| 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 |
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 |
| \[ |
\[ |
| S^*(f,P) := \sum_{R\in P} \mathrm{meas}(R)\sup_{x\in R} f(x),\qquad |
S^*(f,P) := \sum_{R\in P} \mathrm{meas}(R)\sup_{x\in R} f(x),\qquad |
| S_*(f,P) := \sum_{R\in P} \mathrm{meas}(R)\inf_{x\in R} f(x). |
S_*(f,P) := \sum_{R\in P} \mathrm{meas}(R)\inf_{x\in R} f(x). |
| \] |
\] |
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| Polyrectangles are then used to define the Peano Jordan measure of subsets of $\mathbb R^n$ and to define Riemann multiple integrals. |
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. |
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 each of the $n$ intervals defining $R$. |
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 $S$. |
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 $S$. |
