TITLE: How is the Riemann integral a special case of the Stieltjes integral?
QUESTION [12 upvotes]: From Rudin's Principles of mathematical analysis,

6.2 Definition
Let $\alpha$ be a monotonically increasing function on $[a,b]$. ... Corresponding to each partition $P$ of $[a,b]$, we write
$$\Delta \alpha_i = \alpha(x_i) - \alpha(x_{i-1}).$$

He then goes on to define the Riemann Stieltjes integral of $f$ with respect to $\alpha$, over the interval $[a,b]$.
The Riemann integral is then pointed out to be a special case of this when $\alpha(x)=x$.
With $\alpha(x)=x$, I understand $\Delta x = x_i - x_{i-1}$ to represent the directed magnitude of the "base of the approximating rectangle" that we then multiply by the value of $f$ taken somewhere within this interval, thus obtaining the area of an approximating rectangle.
I don't know where to begin to interpret the case where $\alpha(x) \not\equiv x$.

REPLY [2 votes]: In Stieltjes integral you assign different importance to different parts of the set you are integrating over.
The usefulness of this would become clearer when you know the theory of Lebesgue measure, which generalizes this even further. For example, there is the Dirac measure which when used for integrating functions cares only about the value of a function at a point (typically the origin). 
Understanding it in the Stieltjes form is not any harder. Moreover, it will pave the way for measure theory.