measure of the image of the the unit open disc by a holomorphic map

Question

I found the following interesting exercice in a textbook:

Let $f$: $\Bbb E \to \mathbb{C}$ be a holomorphic and injective map ('Schlicht function'), where $\Bbb E=\{z \in \Bbb C:|z|<1\}$. $f(z)=\sum_{n=0}^{\infty} c_{n}z^n$.

While it is clear to me that $f$ is a diffeomorphism from $\Bbb E$ to $f(\Bbb E)$ (which is open) the following is not clear:

If $\sum_{n=0}^{\infty} n|c_{n}|^2<\infty$ then $f(\Bbb E)$ is measurable (as a Borel subset of $\mathbb{C}$) and $\lambda_{2}(f(\Bbb E))=\pi\cdot\sum_{n=1}^{\infty}n|c_{n}|^2$. ($\lambda_{2}$ for the Lebsegue measure on $\mathbb{C}$ )

If you could give me a hint to show the measurability of the set $f(\Bbb E)$. For the formula I use the transformation formula, I have not yet finished but it should work.

Answer 1

The Open mapping theorem in complex analysis states that every nonconstant holomorphic function sends open sets to open sets. Therefore $f(E)$ is open. Every open set is Borel measurable.

In case you're still working on the formula for area, the rest is spoilered.

The Jacobian of $f$ is $|f'|^2=f' \overline{f'}$. Multiply the series for $f'$ by its conjugate, and integrate with respect to polar angle. Most of the terms will integrate to $0$.