Vtyjkyuk

In my elementary analysis courses the Riemann integral was introduced by constructing the Darboux integral and in the special case of continuous functions equivalence was shown. I've now stumbled upon something called a Riemann-Stieltjes integral and I'm curious if it's possible to construct this integral in a similar manner to the Darboux integral and then proving that they're equivalent that way. Is this possible? I've though about proving this myself but if it's already known that it's not possible it would be nice to know so that I don't waste my time.

Riemann integrability is the same as Darboux integrability. $f$ is Riemann or Darboux integrable on $[a,b]$ iff the set of discontinuities of $f$ is of "measure zero." This "measure zero" criteria was discovered by Lebesgue, and was what motivated Lebesgue's definition of a general measure of a set of real numbers, and his new integral.

Now suppose that $g$ is a real non-decreasing function on $[a,b]$ and that $f$ is a bounded real function on $[a,b]$. The Darboux-Stieltjes integral may exist when the Riemann-Stieltjes does not because of issues concerning the behavior of $f$ near the discontinuities of $g$. However, if $f$ is continuous at every point of discontinuity of $g$, then $f$ is Riemann-Stieltjes integrable with respect to $g$ iff $f$ is Darboux-Stieltjes integrable with respect to $g$, which is why the issue does not come up when $g(x)=x$.

This is in fact possible. See for example https://en.wikipedia.org/wiki/Riemann%E2%80%93Stieltjes_integral .