Cartan's lemma (potential theory) explained

In potential theory, a branch of mathematics, Cartan's lemma, named after Henri Cartan, is a bound on the measure and complexity of the set on which a logarithmic Newtonian potential is small.

Statement of the lemma

The following statement can be found in Levin's book.^[1]

Let μ be a finite positive Borel measure on the complex plane C with μ(C) = n. Let u(z) be the logarithmic potential of μ:

u(z) = (1)/(2 \pi) \int _{C} log |z - \zeta|d\mu(\zeta)

Given H ∈ (0, 1), there exist discs of radii r_i such that

\sum _{i} r _{i} < 5 H

and

u(z) \ge (n)/(2 \pi) log (H)/(e)

for all z outside the union of these discs.

Notes

1. B.Ya. Levin, Lectures on Entire Functions