I was reading Rudin's, "Principles of Mathematical Analysis", specifically the section about the Riemann Integral and I've ran into some "shaky" notation. Can someone just explain to me geometrically what is going on here (by here I mean what is def $6.2$ saying)?
$6.2$. Definition. Let $\alpha$ be a monotonically increasing function on $[a,b]$ .....he goes on to say, "Corresponding to each partition $P$ of $[a,b]$, we write $\Delta \alpha_{i} = \alpha(x_{i})-\alpha(x_{i-1})$." It is clear that $\Delta\alpha_{i}\geq0$. For any real function $f$ which is bounded on $[a,b]$ we put, $U(P,f,\alpha)=\sum_{i=1}^{n}M_{i}\Delta\alpha_{i}$.
Definitons which were given before this:
$M_i$ is defined as, $M_i=lub\ f(x)$ with $(x_{i-1}\leq x\leq x_i)$. Also, the partition $P$ is given on $[a,b]$ a finite set of points $x_0\leq x_1\leq ...\leq x_{n-1}\leq x_n$ , where $a= x_0\leq x_1\leq ...\leq x_{n-1}\leq x_n=b$
I hope it is clear what I'm asking, and if anyone has the book it is pages $104-105$.
You may think of the following. In Riemann integral we chop the interval $[a, b]$ into smaller intervals $[x_i x_{i+1}]$ and calculate the area of the two rectangles which is $$M_i(x_{i+1} - x_i) \text{ and }m_i(x_{i+1} - x_i)$$
and using these two area to approximate the "area under graph" of $f$.
But when $\alpha$ is given, we are giving the intervals $[a, b]$ another measurement: we define the "length" of $[x_i, x_{i+1}]$ to be $\alpha(x_{i+1} - \alpha(x_i)$ (This is nonnegative as $\alpha$ is increasing). So we can still estimate the "area under graph" with respect to this measurement. By taking limit we come up to the integral
$$\int_a^b f d\alpha\ .$$
We may also think (as Tyler stated) that $\alpha$ is a way to give a weight to each element in the interval. To the extreme, if $\alpha$ is the function
$$\alpha (x) = 0\ \text{ when } x<c \ \text{ and } \alpha(x) = 1 \text{ when }x\geq c\ ,$$
when this is given weight solely to the point $c$ (This is sometimes called the point measure). In this case, one would have
$$\int_a^b f d\alpha = f(c)$$
if $f$ is continuous at $c$.