Comparison Principle for Extremal Length

Theorem
Let $X$ be a Riemann surface.

Let $\Gamma_1$ and $\Gamma_2$ be families of rectifiable curves (or, more generally, families of unions of rectifiable curves) on $X$.

Let every element of $\Gamma_1$ contain some element of $\Gamma_2$.

Then the extremal lengths of $\Gamma_1$ and $\Gamma_2$ are related by:
 * $\lambda \left({\Gamma_1}\right) \ge \lambda \left({\Gamma_2}\right)$

More precisely, for every conformal metric $\rho$ as in the definition of extremal length, we have:
 * $L \left({\Gamma_1, \rho}\right) \ge L \left({\Gamma_2, \rho}\right)$

Proof
We have:

This proves the second claim.

The second claim implies the first by definition.