# Connection between Riemann and Riemann-Stieltjes Integrals

Let functions $f$ and $h$ be Riemann integrable and $H(x) = \int_a^x h(t)dt$. Is it always true that the Riemann-Stieltjes integral $\int_a^bf(x)dH(x)$ exists and $\int_a^b f(x) dH(x) = \int_a^bf(x)h(x) dx$?

I remember seeing this used in a reference without a proof. How is it proved?

The closest I could find was the more restrictive Theorem 6.17 in Principles of Mathematical Analysis by Rudin. He proves $\int_a^b fd\alpha = \int_a^bf(x) \alpha'(x) dx$ when $\alpha$ is differentiable in $[a,b]$ and the derivative $\alpha'$ is Riemann integrable.

Theorem: Let $h$ be a real Riemann integrable function on $[a,b]$, and let $f$ be a bounded real function on $[a,b]$. Then $f$ is Riemann-Stieltjes integrable with respect to $H(x)=\int_a^x h(t)dt$ iff $fh$ is Riemann integrable on $[a,b]$. And, in that case, $$\int_{a}^{b}fdH = \int_{a}^{b}fhdx.$$ Proof: If you have an augmented partition $\mathcal{P}$ of $[a,b]$, then $$\sum_{\mathcal{P}}f(t_j^*)\Delta_j H-\sum_{\mathcal{P}}f(t_j^*)h(t_j^*)\Delta_j x = \sum_{\mathcal{P}}f(t_j^*)\int_{t_{j-1}}^{t_j}\{h(t)-h(t_j^*)\}dt. \;\;(*)$$ Let $M$ be a bounded for $f$. Then the right side is bounded absolutely by $$M\sum_{\mathcal{P}}\omega(h,I_j)\Delta_j x \le M( \overline{S}_{\mathcal{P}}(h)-\underline{S}_{\mathcal{P}}(h)).$$ Here $\omega(h,I_j)$ is the oscillation of $h$ over the partition interval $I_j$, and the terms on the right are upper and lower sums. So that means the right side of $(*)$ tends to $0$ as the norm of the partition tends to $0$. Therefore, the limit of one of the sums on the left of $(*)$ exists iff the limit of the other sum on the left exists and, in that case, the two limits are equal. $\;\;\blacksquare$
In general, $H$ as defined is continuous and of bounded variation. Hence, the integral $\int_a^b f \, dH$ exists. Since $H$ has bounded variation, the derivative $h = H'$ exists almost everywhere and the result follows.