How is a set of two dimensions measured by integrating an indicator function? The indicator function $\chi$ of a set $A$ is that function for which
$$\chi_A(x) =
\begin{cases}
1 & \text{if $x\in A$} \\
0 & \text{otherwise}
\end{cases}$$
(Some people prefer $\mathbf{1}_A$ as a symbol instead.)
When dealing with a universal set $\Bbb U$ (usually $\Bbb R$ I presume) the measure of the set $A\subset\Bbb U$ is, as I understand it,
$$m(A) = \int_\Bbb{U} \chi_A(x) \, dx$$
(where $x$ is simply a dummy variable).
Of course, sets of two dimensions have measures. For example, $m([4,6]\times[0,10])=20$, which I could tell you having taken simple geometry. However, in general terms, how is the measure of a two-dimensional set defined in terms of an indicator function and integration?
Naturally, the element of each set will be an ordered pair, and I anticipate that the measure of the two-dimensional set is not necessarily the product of the measures of each dimension.
 A: The measure of a two-dimensional set can be defined exactly as you have written it --
 if the characteristic, or indicator, function $\chi_A(x)$ is defined to be $1$ over the set $A$ and $0$ elsewhere, the measure of a set $A$ is indeed equal to $$m(A) = \int_\mathbb{U} \chi_A dm$$
This is the notation I learned in Rudin -- instead of integrating with respect to a bound variable x ($dx$) you integrate with respect to the measure $m$. Does this seem circular? 
The thing is -- if I am remembering correctly -- we can resolve this with exactly the sort of simple set you bring up. The measure of a rectangle in $\mathbb{R}^2$, or a box in $\mathbb{R}^n$ (regardless of we include or exclude the vertices/edges/faces/etc.) is defined to be the product in $\mathbb{R}$ of the lengths of the sides. This is useful because every set that can be written as the union of a countable collection of disjoint boxes can be measured by breaking it into boxes, measuring those with the defined measure, and summing. 
By the way, if you're looking for a book to introduce measure theory as it is used in abstract integration, Real and Complex Analysis by Walter Rudin is pretty standard.
