Majority of sources discussing stochastic integral (Ito), begins with a discussion why Riemann-Stieltjes integral fails when integrating with respect to Brownian motion. More precisely, they state that it fails because of unbounded variation of Brownian motion (which makes sense intuitively).
In a book of Robert P. Dobrow, the author defines the Riemann-Stieltjes integral with respect to Brownian motion and gives only technical conditions on the integrand function, but not the Brownian motion.
It seems somehow contradictory to what most of other books say because they usually only mention the unbounded variation of BM.
I actually found another source where this is briefly discussed and it says that bounded variation of an integrator (e.g. BM) is not a necessary condition actually for Riemann-Stieltjes integral to exist.
So is bounded variation really a stronger assumption than necessary for Riemann-Stieltjes integral exist and there are some functions that you can still integrate in a Riemann-Stieltjes sense wrt to Brownian motion?