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.
Am I making sense, so far ?
$$-------------------------------------$$ All I want is to integrate a measurable function in a measurable space over the image of logarithm function, whenever it is possible, instead of measurable sets only.
So I expect some comments and discussion in this direction, focusing on Haar measure mainly.
