Theorem: Extremal Length of Union

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 $X$. 
Then the extremal length of their union satisfies:
:$\dfrac 1 {\map \lambda {\Gamma_1 \cup \Gamma_2} } \le \dfrac 1 {\map \lambda {\Gamma_1} } + \dfrac 1 {\map \lambda {\Gamma_2} }$
Suppose that additionally $\Gamma_1$ and $\Gamma_2$ are disjoint in the following sense: there exist disjoint Borel subsets:
:$A_1, A_2 \subseteq X$ such that $\ds \bigcup \Gamma_1 \subset A_1$ and $\ds \bigcup \Gamma_2 \subset A_2$
Then
:$\dfrac 1 {\map \lambda {\Gamma_1 \cup \Gamma_2} } = \dfrac 1 {\map \lambda {\Gamma_1} } + \dfrac 1 {\map \lambda {\Gamma_2} }$

Proof:
Set $\Gamma := \Gamma_1\cup \Gamma_2$.
Let $\rho_1$ and $\rho_2$ be conformal metrics as in the definition of extremal length, normalized such that:
:$\map L {\Gamma_1, \rho_1} = \map L {\Gamma_2, \rho_2} = 1$
We define a new metric by:
:$\rho := \map \max {\rho_1, \rho_2}$
{{explain|Prove that $\rho$ is a metric}}
Then:
:$\map L {\Gamma, \rho} \ge 1$
and:
:$\map A \rho \le \map A {\rho_1} + \map A {\rho_2}$
{{explain|What is $A$?}}
Hence:
{{begin-eqn}}
{{eqn | l = \frac 1 {\map \lambda \Gamma}
      | o = \le
      | r = \frac {\map A \rho} {\map L {\Gamma, \rho} }
      | c = 
}}
{{eqn | o = \le
      | r = \map A \rho
      | c = 
}}
{{eqn | o = \le
      | r = \map A {\rho_1} + \map A {\rho_2}
      | c = 
}}
{{eqn | r = \frac 1 {\map L {\Gamma_1, \rho_1} } + \frac 1 {\map L {\Gamma_2, \rho_2} }
      | c = 
}}
{{end-eqn}}
Taking the infimum over all metrics $\rho_1$ and $\rho_2$, the claim follows.
Now suppose that the disjointness assumption holds, and let $\rho$ again be a Borel-measurable conformal metric, normalized such that $\map L {\Gamma, \rho} = 1$. 
We can define $\rho_1$ to be the restriction of $\rho$ to $A_1$, and likewise $\rho_2$ to be the restriction of $\rho$ to $A_2$.
By this we mean that, in local coordinates, $\rho_j$ is given by
:$\map {\rho_j} z \size {\d z} = \begin {cases}
\map \rho z \size {\d z} & : z \in A_j \\
0 \size {\d z} & : \text {otherwise}
\end {cases}$
{{explain|The above section from "By this we mean" needs considerably more explanation, as none of the concepts introduced here can be understood without reference to links from elsewhere.}}
Then:
:$\map A \rho = \map A {\rho_1} + \map A {\rho_2}$
and:
:$\map L {\Gamma_1, \rho_1}, \map L {\Gamma_2, \rho_2} \ge 1$
{{explain|How do these two statements follow from what went before?}}
Hence:
{{begin-eqn}}
{{eqn | l = \map A \rho
      | r = \map A {\rho_1} + \map A {\rho_2}
      | c = 
}}
{{eqn | o = \ge
      | r = \frac {\map A {\rho_1} } {\map L {\Gamma_1, \rho} } + \frac {\map A {\rho_2} } {\map L {\Gamma_2, \rho} }
      | c = 
}}
{{eqn | o = \ge
      | r = \frac 1 {\map \lambda {\Gamma_1} } + \frac 1 {\map \lambda {\Gamma_2} }
      | c = 
}}
{{end-eqn}}
Taking the infimum over all metrics $\rho$, we see that:
:$\dfrac 1 {\map \lambda {\Gamma_1 \cup \Gamma_2} } \ge \dfrac 1 {\map \lambda {\Gamma_1} } + \dfrac 1 {\map \lambda {\Gamma_2} }$
Together with the first part of the Proposition, this proves the claim.
{{qed}}
Category:Geometric Function Theory
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