I am reading Zeros of Gaussian Analytic Functions by Mikhail Sodin and he gives an much-too-easy proof of density of zeros of a Gaussian Analytic function.
Definition A Gaussian analytic function $f(z) = \sum a_n f_n(z)$ where $a_n$ is a Gaussian normal complex random variable and $f_n(z)$ are holomorhpic functions.
Sodin studies the density of the zeros the distribution and gets a proof by Edleman-Kostlan
$$ n_f = \sum_{f(a)= 0} \delta (x-a) = \tfrac{1}{2\pi}\log | \,f \,| $$
Theorem $ \mathbb{E}[n_f]=\tfrac{1}{2\pi}\log || \,f \,|| $ where $||\, f \,||^2 = \sum |\,f_n(z)\,|^2$ the Riesz measure of the norm of $\vec{f}$.
The proof talks about the Fubini study metric and it's just three steps:
$$ \mathbb{E}[n_f] = \frac{1}{2\pi} \mathbb{E} \big[ \Delta \log |\,f\,| \big] = \frac{1}{2\pi} \log ||\,f\,||$$
The last step is an abbreviation for $\mathbb{E}\log |\vec{a}| = \log ||\vec{a}|| $
Why is this equivalent to the Crofton formula which expresses arc-length in terms of the intersection of a curve with random lines?
$$ \int_{\mathrm{Gr}_1(\mathbb{R}^2)} \# \{ \ell \cap C\} d\mu = 2\pi | C|$$
