\begin{definition}[Definition:Extremal Length]
Let $\Gamma$ be a set of rectifiable curves in the complex plane $\C$.
We consider conformal metrics of the form $\map \rho z \cmod {\d z}$, where:
:$\rho: \C \to \hointr 0 \to$ is Borel measurable
and:
:the area $\ds \map A \rho := \iint \rho^2 \paren {x + i y} \rd x \rd y$ is finite and positive.
Every $\gamma \in \Gamma$ has a distance function with respect to such a metric, defined by:
:$\ds \map L {\gamma, \rho} := \int_\gamma \map \rho z \size {\d z}$
We define:
:$\ds \map L {\Gamma, \rho} := \inf_{\gamma \mathop \in \Gamma} \map L {\gamma, \rho}$
and:
:$\ds \map \lambda \Gamma := \sup_\rho \frac {\map L {\Gamma, \rho}^2} {\map A \rho}$
The quantity $\map \lambda \Gamma$ is called the '''extremal length''' of the curve family $\Gamma$.
Its reciprocal:
:$\mod \Gamma := \dfrac 1 {\map \lambda \Gamma}$
is called the '''modulus''' of $\Gamma$.
\end{definition}