I'm reading Rudin's Principles of Mathematical Analysis $6^{th}$ chapter and am wondering why doesn't he require the weaker condition of $\alpha$ (the integrator) being bounded on $[a,b]$ ?
Is he doing that just to simplify things ? Or is there an incompatibility between his definition of the Riemann–Stieltjes integral (he uses the definition with Upper and Lower integrals) and the weaker condition of $\alpha$ being bounded on $[a,b]$ ?
I was thinking of this possibility that maybe the Upper and Lower integrals approach is only compatible with the monotonically increasing integrators $\alpha$.