Let $\alpha$ be nondecreasing on $[0,1]$. Then Riemann-Stieltjes integrable function is well defined with respect to the integrator $\alpha$.
My questions are as follows:
(1) Let $f$ be Riemann-Stieltjes integrable on $[0,1]$ with respect to $\alpha$. Is $f$ Lebesgue integrable on $[0,1]$ with respect to the measure induced by $\alpha$?
(2) Let $f$ be continuous on $[0,1]$. Is $f$ Lebesgue integrable on $[0,1]$ with respect to the measure induced by $\alpha$?
(3) If not, what additional assumptions on $\alpha$ do we need for guaranteeing Lebesgue integrability of $f$? For example, $\alpha$ is right-continuous or continuous on $[0,1]$.
[EDIT] I found a paper related to my question [https://www.jstor.org/stable/2323739?seq=1#metadata_info_tab_contents]
It seems that (1) is true. Am I right?
I would be grateful if you give any comment for my question.