Lebesgue's criterion for Riemann-integrability:
Let $f$ be defined and bounded on $[a,b]$ and let $D$ denote the set of discontinuities of $f$ in $[a,b]$.Then $f\in R$ on $[a,b]$ if,and only if,$D$ has measure zero.
My question is,does the Lebesgue's criterion for Riemann-integrability also hold for Riemann-Stieltjes integral?That is,does the following hold?
Let $f$ be defined and bounded on $[a,b]$ and let $D$ denote the set of discontinuities of $f$ in $[a,b]$.Then $f\in \mathcal{R}(\alpha)$ on $[a,b]$ if,and only if,$D$ has measure zero.($\alpha$ is a monotonically increasing function on $[a,b]$).