I have to compute the Lebesgue measure of certain subsets of $\bar{\Bbb C}$, which is a Riemann Surface.
What I thought is to apply suitable charts to reduce myself to work with subsets of the complex plane, according to the sets I deal with; if such a set does not touch $\infty$ I just use the identity chart, otherwise, if $U\subset\bar{\Bbb C}$ is the open set I have to measure and $a$ is a point NOT belonging to $U$ I use the inversion centered in a $a$, that is $z\mapsto\frac1{z-a}$ and then I take the Lebesgue measure of this set.
This doesn't seem consistent and I suspect there is something I am ignoring to deal with such a problem, like using certain metrics and so on.
Any hint to make this rigorous? Thanks
EDIT Say I have the usual unit disk $\Delta:=\{|z|<1\}$; as a subset of $\Bbb C$ its Lebesgue measure is $2\pi$. From this I can consider $\Delta$ as a subset of $\bar{\Bbb C}$ assigning it the same measure.
Maybe my argument fails when I take the Lebesgue measure of sets of $\Bbb C$ got from $\bar{\Bbb C}$ with different charts. But it can be consistent if I choose a point not belonging to any of the sets I will consider and use always the same chart. Could this work?
Identify $\mathbb{C}\cup\{\infty\}$ with the standard $\mathbb{S}^2$ via stereographic projection. Then the spherical metric $ds=2\lvert dz\rvert/(1+\lvert z\rvert^2)$ induces a measure $\mu$ which is $$\frac{d\mu}{d\lambda_{\mathbb{C}}}(z)=\frac4{(1+\lvert z\rvert^2)^2}$$ where $\lambda_\mathbb{C}$ is the standard Lebesgue measure on $\mathbb{C}\cong\mathbb{R}^2$. This formula remains the same as long as you are doing stereographic projections onto equatorial planes and $z$ is the coordinate on that plane. In particular, on $w=1/z$ chart, it is also $$ \frac{d\mu}{d\lambda_{\mathbb{C}}}(w)=\frac4{(1+\lvert w\rvert^2)^2}. $$
Note the only isometries of the Riemann sphere are rotations of $\mathbb{S}^2$, which is not the full Mobius group.