A question on the definition of the Riemann–Stieltjes integral I have been reading Wikipedia's article on the Riemann–Stieltjes integral (https://en.wikipedia.org/wiki/Riemann%E2%80%93Stieltjes_integral) and I don't understand why the Riemann–Stieltjes integral isn't equaivalent to the Generalized Riemann–Stieltjes integral. I mean, if instead of $g$ we take the identity function (i.e. we consider the case of the Riemann integral) then they are the same. Why does this change for other $g$'s? I am quite confused because my textbook uses the definition Generalized Riemann–Stieltjes integral as the definition for the Riemann–Stieltjes integral and this is why I want to understand the difference between them.
EDIT: I would also like to know which properties of the generalized integral transfer to the ungeneralized one and which do not to fully grasp them.
 A: *

*Pollard, Henry (1920). "The Stieltjes integral and its generalizations". The Quarterly Journal of Pure and Applied Mathematics. 49.
Pollard's paper appears to be where the generalization was introduced. The paper begins on page 73 and that page has only the title, the author's name and affiliation, and the references. The link above is to page 74.
On page 80, Pollard defines two functions:
$$
f(x) = \begin{cases} 0 & x < 1, \\ k & x\ge 1, \end{cases}
$$
$$
\varphi(x) = \begin{cases} 0 & x\le 1, \\ 1 & x>1. \end{cases}
$$
He seems to claim that as a generalized Riemann–Stieltjes integral, apparently defined as a Moore–Smith limit of a net indexed by partitions of the interval $[0,2],$ the integral
$$
\int_0^2 f(x)\,d\varphi(x)
$$
exists and is equal to $k(\varphi(2) - \varphi(1)),$ but that as a Riemann–Stieltjes integral, defined as a limit as the mesh of the partition approaches $0,$ that integral does not exist. Pollard calls the the generalized Riemann–Stieltjes integral "the modified Stieltjes integral."
But I haven't carefully sifted through all the details.
