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When does the measure integral of the form $\int_{\log(S)} f d \mu$ exist ?

Here $\mu$ can be any measure (Lebesgue, Borel, Haar etc), $f$ is a measurable function, $S$ is any measurable set with respect to $\mu$.

How many ways can we define measure integral on the images of logarithm (real or complex or any number field) ?

For example, if $f$ is a real valued measurable function on the measurable set $[1,b], b \in \mathbb{R}$, then we have a Lebesgue measure $\mu$ integrating $f$ over the set $[0, \log(b)]$ as follows: $$I=\int_{\log([1,b])} f d \mu=\int_{\left[0,\log(b)\right]}f d \mu.$$ In the above case, $[0, \log(b)]$ is locally compact Housdorff, so it is a Borel set and we can consider Haar measure to define the above integral.

Now in the complex number case, Assume a closed contour $S:~1 \leq |z| \leq b$, $b \in \mathbb{R}$ and $z$ is a complex number. The principal branch of complex logarithm $\log(z)$ takes $S$ to a rectangular region $R:~[0, \ln b] \times (-\pi, \pi]$ (not good at complex, at least assume), then this image $R=\log(S)$ is measurable. So we can get complex valued measurable function $f$ and integrate it over $\log(S)$ as follows: $$I=\int_{\log(S)}f d \mu=\int_Rf d \mu,$$ where $\mu$ is the complex measure.

I want to define a measure integral over the image of logarithm function, whenever it is possible, instead of measurable sets only.

Any discussion is welcome.

Edit: I just want some examples of measure integrals of the form $\int_{\log(E)} (\text{some function}) \cdot d \mu$, where $\mu$ is some measure, say, Haar measure and $E$ is some subset of the domain of $\log$.

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    $\begingroup$ Can you explain why you do not work with the associated Riemann surface $S$ of $\log$ with the corresponding universal cover $\pi : \mathbb{C} \setminus \{0\} \to \mathbb{C}\setminus \{0\}$ and then integrate over any $\mu'$ with ${\mu'} ^ \pi = \mu$? $\endgroup$ Sep 20 at 17:42
  • $\begingroup$ @DieterKadelka, can you please extend your comment into an answer ? I need to understand it clearly. Thanks $\endgroup$ Sep 20 at 18:49
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For completeness let $R$ be the associated Riemann surface of $\log$ and $\pi : R \to \mathbb{C}^*$ (with $\mathbb{C}^* := \mathbb{C} \setminus \{0\}$) be the corresponding universal cover (see f.i. https://en.wikipedia.org/wiki/Complex_logarithm). Then $\pi$ is surjective, continuous and each (Radon-) measure $\mu$ on $\mathbb{C}^*$ can be written in the form $\mu(A) = \mu'(\pi^{-1}(A))$ for some Radon measure $\mu'$ on $R$. (Each such $\mu'$ leads to the same integral.) And then we can take $$\int_{\mathbb{C}^*} f ~ d\mu := \int_R f \circ \pi d\mu'.$$ I'm not sure that this is what you want.

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  • $\begingroup$ Thank you for your answer. However, my question is more simple. I just want some examples of measure integrals of the form $\int_{\log(E)} (\text{some function}) \cdot d \mu$, where $\mu$ is some measure, say, Haar measure and $E$ is some set in the domain of $\log$ $\endgroup$ Sep 21 at 6:18

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