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 <a href="http://en.wikipedia.org/wiki/Riemann%E2%80%93Stieltjes_integral">Riemann-Stieltjes integral</a> of f(x)?

Perhaps it would be related to the $\Delta$-derivative in <a href="http://en.wikipedia.org/wiki/Time_scale_calculus">time scale calculus</a>.

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 pre-cursor 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 <a href="http://en.wikipedia.org/wiki/Quantum_calculus">quantum calculus</a> can be related to the RS integral (page 7 of <a href="http://www.mat.uc.pt/preprints/ps/p0432.pdf">paper by Abreau</a>), 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 <a href="http://en.wikipedia.org/wiki/Radon-Nikodym_derivative">Radon-Nikodym derivative</a> of the <a href="http://en.wikipedia.org/wiki/Lebesgue%E2%80%93Stieltjes_integration">Lebesgue–Stieltjes integral</a>.