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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.2012-03-10
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    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?2012-03-10
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    So this is a Stieltjes integral?2012-03-10
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    @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.2012-03-10
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    @draks : I think what your looking for are [vector measures](http://en.wikipedia.org/wiki/Vector_measure), 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.2012-03-12
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    @ineff Thanks it does help. What do you think about the subsitutions in 3.?2012-03-15
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    Not "too basic" but "incomprehensible". I don't know what $\int_C f(t) d\mu(at)$ means. Perhaps you could write down an explicit example for us.2012-03-20
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    @GEdgar The example I'm interested is: $\int_C 1 d(P(at))$, where $P(\cdot)$ is the [PrimeZeta function](http://en.wikipedia.org/wiki/Prime_zeta_function) and $C$ could be a curve around several non-trivial roots of $\zeta$.2012-03-20
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    Still don't know what $\mu(at)$ is. You seem to believe that $\mu$ is a function, it is not, at least not acting on numbers. The notation $\int f(t)d\mu(t)$ is just that, a notation, hence you need to explain the (highly non canonical) notation $\int f(t)d\mu(at)$.2012-03-20
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    @DidierPiau, [here](http://math.stackexchange.com/a/49434/19341) the author used $\int_{2}^{x}t^{-s}d\left(\text{li}(t)\right)$, and I was curious, how I can use that. Maybe you could be so kind and explain the differences to me?2012-03-20
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    When $f$ is a regular function, $\int_x^yg(t)df(t)=\int_x^yg(t)f'(t)dt$. What do **YOU** call $\int_x^yg(t)df(at)$? Is it $\int_x^yg(t)dh(t)$ in the preceding sense with $h:t\mapsto f(at)$ or what?2012-03-20
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    @DidierPiau Yes, a scaled version of $f(t)$. Maybe it was misleading to choose $\mu$ instead of anything else. Sorry for the confusion.2012-03-20
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    @DidierPiau, so you say, that I can use $\int_C g(t)d(f(at))=\int_C g(t)f'(at)dt$, given my function $f$ is regular (holomorph)?2012-03-22
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    No I did not say that. And you did not answer my question: what do you call $\int_x^yg(t)df(at)$?2012-03-23
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    @DidierPiau, I did. I agreed with your proposal.2012-03-23
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    Then the result is not $\int g(t)f'(at)dt$... :-)2012-03-23
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    @DidierPiau :-(2012-03-23

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