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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).

Thanks for your help...

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$\mu$ 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
So this is a Stieltjes integral? – GEdgar Mar 10 '12 at 22:13
@GEdgar What do I need to qualify this as a Stieltjes integral? I want to use these things somewhere else, but I'm not sure if they are valid. – draks ... Mar 10 '12 at 22:35
@draks : I think what your looking for are vector measures, with that you can define complex measures and so complex integrals. Vector measures can be simply seen as vector of measures, so to prove property in this context you have to work with the components of the vector measure itselfs. Hope this help. – Giorgio Mossa Mar 12 '12 at 11:01

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