Are increasing functions Riemann-Stieltjes integrable?

Question

It is easy to prove that if $f$ is increasing on $[a,b]$ then it is Riemann integrable.

If $f$ is increasing is it necessarily Riemann-Stieltjes integrable where $\alpha$ is increasing on $[a,b]$?

Answer 1

Take $f=g$ defined on $[0,2]$ by

$$f(x)=\begin{cases} 0 & \text{for }0 \le x <1 \\ 1 & \text{for } 1 \le x \le 2 \end{cases}$$

$f,g$ are increasing. However $\int_0^2 f dg$ doesn’t exists. The lower RS sum is equal to $0$ while the upper one is equal to $1$.

Answer 2

Not necessarily.

It is a theorem in Baby Rudin (Principles of Mathematical Analysis, Theorem 6.9 on page 126) that

If $f$ is monotone on [a,b] and $\alpha$ is continusous (and increasing) on $[a,b]$, then $f$ is Riemann-Stieltjes integrable with respect to $\alpha$

However, if $\alpha$ is not continuous, then $f$ might not be integrable with respect to $\alpha$. Indeed, we have another theorem (6.10 on the same page) which says, roughly, that whenever $\alpha$ and $f$ have different points of discontinuity, $f$ will be integrable with respect to $\alpha$.

As a (potentially fun) exercise, try to find a function $\alpha$ and an increasing function $f$ so that $f$ is not integrable with respect to $\alpha$. Keep in mind they'll need to be discontinuous at the same point!