Understanding a measure of special polygon Can someone explain what the first line on the right page Book Page means in the book by Frank Jones - Lebesgue Integration on Euclidean Space book
"For each $ i \in [1,n]$ let $c_i^1 < c_i^2 < c_i^3 < \ldots < c_i^{m_i}$ be the list of all distinct numbers among $a_i^k$ and $b_i^k$ for $i \leq k \leq N$."
What does $m_i$ mean here? Also is the set of $c_i^1 \ldots$ countable or can it be uncountable?
I have uploaded an image here Book page 
You can take a peek into the book on page 60 here Google books link
Anyone? I really don't want to skip the section since the exercises in this section are based on this proof. I just need to understand how the $I^L$'s are constructed in the proof, visually it makes sense.
 A: Assume we have an $n$-dimensional polygon which is the union of a finite number $N$ of rectangles.  We want to show that we can write this polygon as a union of a finite number of nonoverlapping rectangles.  I think that it is best demonstrated by example.
Let $N = 2$ and $n=2$, assume that your polygon is the union of the two rectangles:
\begin{align*}
I^1 &= [0,3] \times [0,1] \\
I^2 & = [1,2] \times [0,3].
\end{align*}
Note that these rectangles overlap.  Now for $i=1$, we look only at the first factor of each rectangle (i.e. the left side of each product above), and we make a list of all of the distinct numbers we see.  Thus
$$m_i \mbox{ is the (finite) number of distinct numbers you see in column } i \mbox{ of your product.}$$
Here $m_1 = 4$ and $m_2 = 3$, and we have:
\begin{align*}
i = 1: \ \ \ c_1^1 & = 0, \ \ \ c_1^2 = 1, \ \ \ c_1^3 = 2, \ \ \ c_1^4 = 3 \\
i=2: \ \ \ c_2^1 & = 0, \ \ \ c_2^2 = 1, \ \ \ c_2^3 = 3.
\end{align*}
Now $\mathscr{L}$ is any $n$-tuple we can obtain by choosing one position (except the final position) for each $i$.  Thus from the first row above we can choose position $1$, $2$, or $3$, and from the second we can choose position $1$ or $2$.  This gives six options for $\mathscr{L}$:
\begin{align*}
(1,1), \ (1,2), \ (2,1), \ (2,2), \ (3,1) \ (3,2).
\end{align*}
Finally we create the rectangle corresponding to each of these options:
\begin{align*}
I^{(1,1)} & = [0,1] \times [0,1] \\
I^{(1,2)} & = [0,1] \times [1,3] \\
I^{(2,1)} & = [1,2] \times [0,1] \\
I^{(2,2)} & = [1,2] \times [1,3] \\
I^{(3,1)} & = [2,3] \times [0,1] \\
I^{(3,2)} & = [2,3] \times [1,3]
\end{align*}
Note that these rectangles are nonoverlapping and will cover the polygon, and note that the polygon is actually equal to the union of the first, third, fourth, and fifth polygons in the list above.
Overall, the idea is to use your finite number of rectangles and create a kind of discrete lattice by breaking the polygons up into the smallest possible pieces, as shown in the picture.
