Can you define a "derivative" operator such that its antiderivative F(x) of f(x) can be used in the sense of F(b)-F(a) to calculate the Riemann–Stieltjes integral of f(x)?
Perhaps it would be related to the $\Delta$-derivative in time scale calculus.
This question was motivated by an edit to that wikipedia article which said that the ideas of unifying sums and integrals go back to the idea of the Riemann–Stieltjes integral. Now I'm not sure it's correct to say the RS-integral is a precursor of time-scale calculus as the starting point of time-scale calculus is the derivative and the unification of difference and differential equations, but integrals on other time-scales such as the q-integral in quantum calculus can be related to the RS integral (page 7 of a paper by Abreu), so maybe definite integrals on all time scales can be written in the form of RS-integrals for some suitable choice of step function depending on the time-scale.
Edit: About the same time, it seems, as I asked this question, a new post appeared on the wikipedia discussion page for "Time scale calculus" which suggests the Hilger derivative (delta-derivative) is the same as the Radon–Nikodym derivative of the Lebesgue–Stieltjes integral.
Best Answer
For a Riemann-Stieltjes integral $$\int_a ^b f(x) d \phi(x)$$ the corresponding derivative should be $$f^{(\phi)}(x)=\lim_{y\to x} \frac{f(y)-f(x)}{\phi(y)-\phi(x)}$$ This works at least if $\phi\in C^1$ and $\phi'$ is never zero, that is, in this case we have $$\int_a ^b f^{(\phi)}(x) d \phi(x)=f(b)-f(a)$$