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. 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 π ∫ C log | z − ζ | d μ ( ζ ) {\displaystyle u(z)={\frac {1}{2\pi }}\int _{\mathbf {C} }\log |z-\zeta |\,d\mu (\zeta )}
Given H ∈ (0, 1), there exist discs of radii ri such that
∑ i r i < 5 H {\displaystyle \sum _{i}r_{i}<5H}
and
u ( z ) ≥ n 2 π log H e {\displaystyle u(z)\geq {\frac {n}{2\pi }}\log {\frac {H}{e}}}
for all z outside the union of these discs.
Notes