Understanding Stieltjes-Riemann From my understanding from lectures, the Stieltjes-Riemann integral is a generalization of the Riemann integral. When using the identity function as integrator, the Riemann sum and Stieltjes-Riemann sum are identical. For any other (monotonically increasing) weighting function, the Riemann sum and Stieltjes-Riemann sum may converge to different values for a given interval.
How would I interpret the result of $\int_a^b{f \text{d}\alpha}$ as it does not seem to relate to the area of the graph of $f$ anymore? How would I choose a weighting function $\alpha$ for meaningful results?
 A: The Riemann Integral is just the limit of a riemann sum over a partition indexed by the variable $i$. The Riemann sum looks, like
$$\sum_{i=1}^n f(x_{i-1}) \ \big(id(x_{i})-id(x_{i-1})\big)$$
Of course, this correpsonds with the definition of a Stieltjes integral for $\alpha=id$.
Each summand of the Riemann sum, namely
$$f(x_{i-1}) \ \big(id(x_{i})-id(x_{i-1})\big)$$
corresponds to one of the red rectangles in the following picture:

Since $\alpha$ is monotonically increasing, you find for each $i$, that
$$\alpha(x_{i})-\alpha(x_{i-1})\ge 0$$
Therefore you find for each $i$ a positive real number $c_i$, such that
$$\alpha(x_{i})-\alpha(x_{i-1}) = c_i \big( id(x_{i})- id(x_{i-1}) \big)$$
And finally you can compare just the summands in the Riemann and Stieltjes sum:
$$f(x_{i-1})\ \big(\alpha(x_{i})-\alpha(x_{i-1})\big) = c_i\  f(x_{i-1})\  \big( id(x_{i})- id(x_{i-1}) \big)$$
Notice, that this is just a summand of the the Riemann sum multiplied by the factor $c_i$.
Therefore you can think of a generalized picture in that sense, that $\alpha$ stretches or narrows each red rectangle according to the factor $c_i$. Note, that $c_i$ won't change a "upward pointing" rectangle into a "downward pointing" one, because $c_i>0$.
For example if $c_i=2$ for every $i$, then each of the rectangle will be stretched, such that they are double the size.
If some specific $c_i=0.5$ and every other $c_i=1$, only this specific rectangle will be narrowed down to half of its size.
