# Riemann-Stieltjes integrals

If $f$ is bounded on $[a,b]$, can one define a Riemann-Stieltjes integral

$$\int_a^b f(x) d\alpha(x)$$

when the function $\alpha(x)$ is not monotonically increasing on $[a,b]$? Rudin only seems to define R-S integrals with respect to monotonically increasing functions, but there are sources I've found on the Internet that seem to imply this requirement is optional (some of them have made noises about $\alpha$ only needing to be a function of bounded variation)...what are the bare minimum requirements on $\alpha$ for the above integral to make sense?

Also, when someone talks about a function being of "finite variation", is this the same as saying the function is of bounded variation?

(EDIT: I think I have confirmed that this is true...saying that a function is of finite variation is the same as saying its total variation is finite, and that occurs iff it is of bounded variation.)

Last edited:
mathman
Bounded variation is sufficient, since any such function can easily be represented as the sum of two functions, monotone increasing plus monotone decreasing.