Riemann–Stieltjes integral

Remarks

Example: if $f$ is continuous and $g$ is continuously differentiable on $[a, b]$, then $\int_a^b f \, dg = \int_a^b f(x) g'(x)\, dx$. By the mean value theorem, $g(x_i) - g(x_{i-1}) = g'(d_i)(x_i - x_{i-1})$ for some $d_i \in (x_{i-1}, x_i)$. Uniform continuity of $g'$ lets one replace $d_i$ by $c_i$ with a small error, which leaves a Riemann sum of $f g'$. With $g(x) = x$ the definition is that of the Riemann integral.

Non-example: on $[-1, 1]$ let $f = g$ be $0$ on $[-1, 0]$ and $1$ on $(0, 1]$. If $0 = x_k$ is a partition point, the only nonzero term is $f(c_{k+1})$. It is $0$ for the tag $c_{k+1} = 0$ and $1$ for any other tag. So sums of every small mesh take both values $0$ and $1$, and $f$ is not integrable with respect to $g$.

If $f$ is continuous and $g$ has bounded variation, then $\int_a^b f \, dg$ exists. Some authors use the limit over refinements of partitions instead of the mesh, which gives a slightly larger class of integrable pairs.