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Let f,g be real valued Riemann integrable functions on [a,b] then does it imply, f is Riemann Stieltjes integrable with respect to g on [a,b]? If not please do provide a counter example demonstrating the same.

What if instead of both being integrable one is integrable and the other of bounded variation?

Above are a few questions which I got stuck at while I was trying to characterize the classes of functions which are Riemann stieltjes integrable with respect to an integrator and also those of which are Riemann integrable. Any help or suggestion is appreciated.

Swag Jensen
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  • This is related. As you know, even for the simplest integrating function $g(x)=x$ Riemann and Riemann-Stieltjes are the same and it is not easy to characterize all Riemann integrable functions $f$ other than by Riemann's original definition. I am not sure if your question leads to a fruitful exercise. Also: why Riemann and not Lebesgue? – Kurt G. Mar 12 '23 at 10:30
  • @KurtG. Well...the characterization part perhaps isn't that important.....what am really looking for are counterexample(s).... – Swag Jensen Mar 12 '23 at 11:56
  • @KurtG. Can you please explain how is my question and the one you link related?...I don't see any... – Swag Jensen Mar 12 '23 at 11:59
  • I just took a closer look at the link. The answer there is pretty bad. So forget that. – Kurt G. Mar 12 '23 at 12:09

1 Answers1

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A counterexample is any Riemann integrable function $f:[a,b] \mapsto \mathbb{R}$ which is discontinuous at some point in the interval and $g = f$.

It is proved here that the Riemann-Stieltjes integral of $f$ with respect to $g$ does not exist if the functions are both discontinuous from the left at a point or both discontinuous from the right. (It is possible for the integral to exist, however, if they are both discontinuous only from different directions.)

Hence, if $f$ is Riemann integrable but discontinuous in any way at a point in $[a,b]$, then -- when $f$ is both integrand and integrator -- a common discontinuity from one side is always shared and $\int_a^b f \, df$ fails to exist.

RRL
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