I'm trying to understand contour integrals $I(f, \Gamma):=\int_\Gamma f(\gamma)d\gamma$ in complex analysis and functional analysis as rigorously as possible. It seems to me this has natural resemblance to the Riemann-Stieltjes integrals $\int_a^b fdg$. But in that context, its measure-theoretic analogue, the Lebesgue-Stieljes integral, interprets function $g$, which is required to be of bounded variation, as a $\mathbb{R}$-valued measure.
So in analogy, in general, it'd be ideal if we can define a vector-valued differentiation concept: suppose $\gamma: [0,1]\to A$ is a rectifiable curve, where $A$ is a finite-dim $\mathbb{R}$-algebra, we need to come up with a set function $\gamma_\Sigma: \Sigma_{[0,1]}\to A$, induced by this $\gamma$, so that $\lim_{|E_t|\to 0}\frac{\gamma_\Sigma (E_t)}{|E_t|}$ is defined as the density if this limit exists almost everywhere, where $E_t$ is a measurable set containing $t\in [0,1]$. If $\gamma$ is assumed to be a smooth curve, we should then expect this density to exist and equal the usual derivative $\gamma'(t)$. But then this seems to suggest also generalizing absolute continuity in this vector-measure context as well.
Then all of this should apply to when $A=\mathbb{C}$.
Are there textbooks out there talking about these things? Thanks in advance.