# Basic Questions on Evaluating $\int_C g(t) d(f(t)+f(at))$

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:

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

2. 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?

3. 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).

$\mu$ is what? Measure? Defined on the contour? So $\mu(at)$ is another measure? Defined on the same contour? How? 1. This is how I would define "addition of measures" I guess. – GEdgar Mar 10 '12 at 20:01
Which conditions does $\mu$ have to fuflfill to be put into $\int f d(\mu)$? In my case $\mu$ is a complex function. Is this a problem? – draks ... Mar 10 '12 at 22:01