Suppose the following complex integral over a countour $C$: $$ \int_C g(t) d(f(t)+f(at)), $$ where $f(t)$ is a complex-valued function. My questions are:

Is this possible at all? Definition of Lebesgue–Stieltjes says, that I need a $f$ that is real, so this wouldn't apply.

When can one split it like $\int_C g(t) d(f(t))+\int_C g(t) d(f(at))$? Is this the "addition of measures" as @GEdgar guesses in his comment and as mentioned here?

How can one use the substitution $u=at$ for the second integral?

- Will this give something like $a\int_{C'} g(u/a) d(f(u))$, where $C'$ is a scaled version of $C$?
- I can also imagine substituting $d(f(at))=\frac{d(f(at))}{d(f(t))}d(f(t))=\frac{d(f(at))/dt}{d(f(t))/dt}d(f(t))$, but how to deal with $g$ then...? What do I substitute there?
- The case $d(f(t^a))$ would also be interesting...Can I treat it the same way?

**EDIT** If the question is too basic, a reference would also be fine. Otherwise feel free to give partial **answers** (on comments I just can help you to a Pundit badge).

Thanks for your help...