\begin{lemma}\label{lem:stieltjes_ibp_Ioc}
Let $a \le b$ and let $g : \mathrm{StieltjesFunction}(\mathbb R)$.
Let $f : \mathbb R \to \mathbb R$ be absolutely continuous on $[a,b]$.
Assume $f'$ is integrable on $(a,b]$ and $x \mapsto g(x) f'(x)$ is integrable on $(a,b]$.
Then
\[
\int_{(a,b]} f(x)\, d(g.\mathrm{measure})(x)
= f(b)g(b) - f(a)g(a) - \int_{(a,b]} g(x) f'(x)\,dx .
\]
\end{lemma}