By defining the Riemann–Stieltjes integral, Apostol in Mathematical analysis (1974), require for the integrand and integrator to be bounded on the definition interval, but I do not see anywhere that he requires that the integrator (integrating function) be monotone.
Practically everywhere else where a looked, this assumption is given. Proves are in general easier with monotonicity.
- I'm correct that Apostol does not require monotonicity?
- What are the consequences if we introduce monotonicity? Do we lose a lot?