Definition of Riemann-Stieltjes integration
Let $\alpha:[a,b]\rightarrow \mathbb{R}$ be a monotonically increasing function and $f:[a,b]\rightarrow \mathbb{R}$ be a bounded function.
Then, $f$ is RS-integrable iff the supremum of the Darboux upper sum $U(P,f,\alpha)$ and the infimum of the Darboux lower sum $L(P,f,\alpha)$ coincides where the supremum and infimum ranges over partions $P$ of $[a,b]$. In this case, the conincident value is denoted by $\int_a^b f d\alpha$.
As written in wikipedia, this definition can be extended to bounded variation parameter as follows:
Let $g:[a,b]\rightarrow \mathbb{R}$ be of bounded variation. Define $g_1(x)= V_a^x (g)$ (the total variation and $g_2(x)= g_1(x) - g(x)$. Then, $g_1$ and $g_2$ are monotonically increasing. Thus, a bounded function $f$ is said to be integrable along $g$ iff $f$ is RS-integerable along $g_1$ and $g_2$, and $\int fdg$ is defined as $\int fdg_1 - \int f dg_2$.
This works fine. However, in analysis, we need more than this. When $g$ is a complex-valued function of bounded variation, how do I extend the above definitions?
(1) One approach to define integral of functions along a complex valued bounded variation is to have $\lim_{\text{mesh(P)}\to 0} | I - \sum_{i=1}^n f(\tau_i)[g(t_i)-g(t_{i-1})]| =0$. However, this definition is stronger than the above definitions in the case $g$ is monotone or real-valued bounded variation.
How do I extend $g$ to a complex-valued functions?
EDIT:
Let's define $\int f dg$ as $\int Re(f) d Re(g) - \int Im(f) d Im(g) + i (\int Re(f) d Im(g) + \int Im(f) d Re(g)$. Then, when $f$ is continuous, would the above definition (1) coincide with this one?
If $g \colon [a,b]\to \mathbb{C}$ is a function of bounded variation, then $h = \operatorname{Re} \circ g$ and $k = \operatorname{Im} \circ g$ are functions $[a,b] \to \mathbb{R}$ of bounded variation. Then one defines that $f$ is RS-integrable with respect to $g$ if and only if it is RS-integrable with respect to $h$ and with respect to $k$, and the Riemann-Stieltjes integral of $f$ with respect to $g$ is defined as
$$\int_a^b f\,dg := \int_a^b f\,dh + i\int_a^b f\,dk.$$