Stieltjes Integral meaning. Can anybody give a geometrical interpretation of the Stieltjes integral:
$$\int_a^bf(\xi)\,d\alpha(\xi)$$
How would we calculate?
$$\int_a^b \xi^3\,d\alpha(\xi)$$
for example.

$$\int_a^b \xi^3\,d\alpha(\xi)=\int_a^b\xi^3a'(\xi)d\mu(\xi) $$ 
in case that $a$ is differentiable. Also $\mu(\xi) $ indicates the Legesgue measure.
In order to find the above formula, you have to use Radon-Nikodym derivatives. (http://en.wikipedia.org/wiki/Radon%E2%80%93Nikodym_theorem).
Generally the function $a$ gives you a way to measure elementary sets, such as intervals. This function will produce the corresponding Stieltjes  measure. In the same way, measuring intervals(or cubes) with the natural way ($\mu([0,1])=1)$, produces Lebesgue measure.
