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

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 $$\log$$.

• 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 ${\mu'} ^ \pi = \mu$? Sep 20 at 17:42
• @DieterKadelka, can you please extend your comment into an answer ? I need to understand it clearly. Thanks Sep 20 at 18:49

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.
• 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 $\log$ Sep 21 at 6:18