Two function not Riemann Stieltjes integrable

Question

I am looking for a simple example without delving too deep into analysis, of two functions $f,g$ on an interval $[a,b]$ for the Riemann Stieltjes integral $\int fdg$ taken over $[a,b]$ does not exist

Answer 1

There is an easy example that meets the wording of your question, namely $f(x)$ is the rationals indicator function and $g(x)$ the identity function: $$ f(x) = \left\{\begin{array}{cl} 1 & x \in \Bbb Q \\ 0 & x \not\in \Bbb Q\end{array}\right.\\g(x) = x $$ Since the indicator function is not Reimann integrable, the (Riemann-Stieltjes) integral $\int f(x)\,dg(x) = \int f\,dx$ does not exist.

(The main reason is that no matter how finely you slice the real line, the min and max of the function within each slice will differ by $1$.)

A much more interesting sort of question is to find example of two functions $f$ and $g$ which are each Riemann integrable, such that (RS) $\int f\,dg$ does not exist, or to find two non-integrable functions $f$ and $g$ such that $\int f\,dg$ does exist.

My hunch is that you can prove that if $f$ and $g$ are each Riemann integrable, then $\int f\,dg$ must exist, and that you can find two non-integrable functions $f$ and $g$ such that $\int f\,dg$ does exist.