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{\bf Question}

Show that $c_*(S) \leq C^*(S)$.



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{\bf Answer}

Let $S$ be a bounded set.  Suppose there are systems of disjoint
rectangles contained within $S$.  Otherwise $c_*(S)=0$ and the
result is trivial.

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{\bf Lemma}  If a rectangle $R$ is decomposed by a grid into
rectangles $R_1, \cdots R_n$

$|R|=\sum|R_n|$

{\bf Proof}  Let the original rectangle be defined by $\{{\bf
x}:a_i\leq x_i\leq b_i\}$.

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Each axis $x_i$ is subdivided by points $x_{i_1}, \cdots
x_{i_{n_i}}$, so that

$a_i=x_{i_1}<x_{i_2}<\cdots<x_{i_{n_i}}=b$

Thus we have the system of rectangles defined by $\{{\bf
x}:x_{ij}\leq x_i\leq x_{ij+1}\},$

$i=1,\cdots n, \hspace{0.1in} j=1,\cdots n_i-1$

$\ds |R|=\Pi_{i=1}^n(b_i-a_i)$

$\ds =\Pi_{i=1}^n(x_{i_{n_i}}-x_{i_1})$

$\ds =\Pi_{i=1}^n \sum_{j=1}^{n_i-1} (x_{ij+1}-x_{ij})$

$\ds =\sum_{j_i=1}^{n_i-1} \Pi_{i=1}^n (x_{ij_i+1}-x_{ij_i})$

$\ds =\sum|R_i|$

Now let $\{R_i\}$ be an arbitrary finite system of rectangles
covering $S$ and let $\{R_i'\}$ be a disjoint system within $S$.
We use all the $a_i, \, b_i, \, a_i', \, b_i'$ of $R_i$ and $R_i'$
to form a grid .  Some of the covering rectangles will contribute
more than once to the grid.  There will be more members of the
grid making up the cover than the linear system.  Therefore $\ds
\sum|R_i|\supseteq \sum|R_i'|$.  Hence the result.



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