Let $G$ be be continuous with bounded variation on finite intervals. If $f$ is continuous then it is well known that $\int_a^bf(G(s))dG(s)=\int_{G(a)}^{G(b)}f(x)dx$. How general can $f$ be so that this formula is valid and where can one find a simple reference (the second question is more important)?
For example, if $G$ is absolutely continuous then it is enough that $f$ is Lebesgue integrable on finite intervals (standard measure theory course material). A similar statement also holds when $G$ is monotone (and continuous). Also, if $f_n\to f$ and $f_n$ are continuous with $|f_n(\cdot)|\le B<\infty$ on $[c,d]$ where $c,d$ are such that $c\le G(s)\le d$ for $a\le s\le b$, then it also holds. This covers quite a few cases.
This surely can be found somewhere. Does anyone know exactly where?