Extremal length and the curve-family modulus of a path family

• Christopher J. Bishop, Quasiconformal Mappings, Ch. 1 §1, printed pp. 1–3. Bishop defines admissibility by ${\mathrm{\ell }}_{\rho }(\mathrm{\Gamma })\ge 1$, modulus by the infimum of $\int {\rho }^2$, and extremal length as the reciprocal modulus; he also states that the density may be taken Borel.
• Mikhail Lyubich, Conformal Geometry and Dynamics of Quadratic Polynomials, vol. I, Ch. 1 §6.1, printed pp. 119–120. Lyubich defines $L_{\rho }(\mathrm{\Gamma })={\mathrm{\ell }}_{\rho }(\mathrm{\Gamma }{)}^2\mathrm{/}{m}_{\rho }(U)$ and takes its supremum over finite-mass metrics; the reciprocal is extremal width and is also the infimum over metrics whose length on every curve is at least $1$.
• Lars Ahlfors and Arne Beurling, Conformal Invariants and Function-Theoretic Null-Sets, §4, printed pp. 114–115. Their supremum convention for extremal length agrees with $\lambda$ below. Their Lemmas 4–5 give the rectangle and round-annulus constants after the curve family is specified.

Assume Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Fix a complex domain $\mathrm{\Omega }\subseteq \mathbb{C}$ (A complex domain is a nonempty connected open subset of $\mathbb{C}$), read as an open subset of the Euclidean plane, with its Borel $\sigma$-algebra (The Borel sigma-algebra of a topological space) and planar Lebesgue area measure (Lebesgue measure is a Radon measure on R^n).

A path in $\mathrm{\Omega }$ is a continuous map $\gamma :[a,b]\to \mathrm{\Omega }$. It is rectifiable when its two coordinate functions have bounded variation (A path in ${\mathbb{R}}^n$ is rectifiable exactly when every coordinate has bounded variation), and $s_{\gamma }$ denotes its arc-length function (The arc-length function $s_{\gamma }(t)=L(\gamma {\mathrm{\mid }}_{[a,t]})$ of a rectifiable path). A path family $\mathrm{\Gamma }$ is a set of paths. When curves joining specified boundary sets of $\mathrm{\Omega }$ are used, allow paths $\gamma :[a,b]\to \bar{\mathrm{\Omega }}$ with $\gamma ((a,b))\subseteq \mathrm{\Omega }$ and endpoints in the named sets; read complex paths as planar rectifiable paths (Complex contours as planar rectifiable paths: the Euclidean, coordinate-BV, and piecewise-C1 dictionaries). Only the interior contributes to length, since each density below is extended by $0$ outside $\mathrm{\Omega }$.

Let $\rho :\mathrm{\Omega }\to [0,+\mathrm{\infty }]$ be Borel measurable with respect to the Borel $\sigma$-algebra of $\mathrm{\Omega }$ (Extended-real-valued measurable functions). Extending it by $0$ on $\mathbb{C}\setminus \mathrm{\Omega }$ gives a Borel function on the plane by the trace identity for subspaces (The Borel sigma-algebra of a subspace is the trace of the ambient Borel sigma-algebra). For a rectifiable path $\gamma :[a,b]\to \bar{\mathrm{\Omega }}$, its arc-length function is continuous and nondecreasing (The arc-length function is continuous and nondecreasing, with increments equal to subpath lengths; it is strictly increasing exactly when no nondegenerate subpath is constant). Extend $s_{\gamma }$ to a continuous nondecreasing function $S_{\gamma }:\mathbb{R}\to \mathbb{R}$ by $S_{\gamma }(t)=0$ for $t\le a$, $S_{\gamma }(t)=s_{\gamma }(t)$ for $a\le t\le b$, and $S_{\gamma }(t)=s_{\gamma }(b)$ for $t\ge b$. Countable Choice gives the associated Lebesgue-Stieltjes Borel measure $d{S}_{\gamma }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), Assuming countable choice, a nondecreasing right-continuous function defines a Borel measure on $\mathbb{R}$). Define the $\rho$-length by ${\mathrm{\ell }}_{\rho }(\gamma ):={\int }_{[a,b]}\rho (\gamma (t))d{S}_{\gamma }(t),$ using the nonnegative Lebesgue integral (The nonnegative Lebesgue integral). This value lies in $[0,+\mathrm{\infty }]$; set ${\mathrm{\ell }}_{\rho }(\gamma )=+\mathrm{\infty }$ when $\gamma$ is not rectifiable. For finite-valued continuous $\rho$ and a rectifiable path whose full trace lies in $\mathrm{\Omega }$, this agrees with the published absolute line integral ${\int }_{\gamma }\rho \mathrm{\mid }dz\mathrm{\mid }$ (The absolute line integral over a rectifiable path using its arc-length function, For a continuous integrand, the Riemann-Stieltjes and Lebesgue-Stieltjes integrals agree, Continuous integrands have complex and absolute line integrals along every rectifiable path). Continuity only on $\mathrm{\Omega }$ does not assert that the zero extension is continuous at boundary endpoints. The continuity of $S_{\gamma }$ makes $d{S}_{\gamma }$ atomless, so changing the integrand on finitely many parameter values does not change the length.

For a path family put ${\mathrm{\ell }}_{\rho }(\mathrm{\Gamma }):={\inf }_{\gamma \in \mathrm{\Gamma }}{\mathrm{\ell }}_{\rho }(\gamma )$, with $\inf \mathrm{\varnothing }:=+\mathrm{\infty }$, and define its area by $A(\rho ):={\int }_{\mathrm{\Omega }}{\rho }^{2}dA\in [0,+\mathrm{\infty }]\mathrm{.}$

The extremal length of $\mathrm{\Gamma }$ is $\lambda (\mathrm{\Gamma }):={\sup }_{\rho }\frac{{\mathrm{\ell }}_{\rho }(\mathrm{\Gamma }{)}^2}{A(\rho )}\in [0,+\mathrm{\infty }],$ where the supremum is over Borel $\rho :\mathrm{\Omega }\to [0,+\mathrm{\infty }]$ with $0<A(\rho )<+\mathrm{\infty }$. A supremum of an empty set is $0$; since $\mathrm{\Omega }$ is a nonempty open domain, it contains a nondegenerate rectangle $Q$, and its indicator has positive finite area (A box in ${\mathbb{R}}^n$ with parameters $a_i\le b_i$ is Lebesgue measurable of measure ${\prod }_{i<n}(b_i-a_i)$, whichever of its faces are included). The value ${\mathrm{\ell }}_{\rho }(\mathrm{\Gamma })=+\mathrm{\infty }$ is allowed. The curve-family modulus is the reciprocal $\mu (\mathrm{\Gamma }):=\frac{1}{\lambda (\mathrm{\Gamma })}\in [0,+\mathrm{\infty }],$ with $1\mathrm{/}0:=+\mathrm{\infty }$ and $1\mathrm{/}+\mathrm{\infty }:=0$. Thus $\lambda (\mathrm{\varnothing })=+\mathrm{\infty }$ and $\mu (\mathrm{\varnothing })=0$, while any family containing a constant path has $\lambda (\mathrm{\Gamma })=0$ and $\mu (\mathrm{\Gamma })=+\mathrm{\infty }$.

Conventions. (1) This library defines extremal length by the displayed supremum and calls its reciprocal the modulus. Sources that define modulus by ${\inf }_{\rho }\int {\rho }^2$ over metrics with ${\mathrm{\ell }}_{\rho }(\mathrm{\Gamma })\ge 1$ call $\mu (\mathrm{\Gamma })$ the modulus and $\lambda (\mathrm{\Gamma })$ the extremal length. (2) Extending $\rho$ by zero outside $\mathrm{\Omega }$ makes densities supported in $\mathrm{\Omega }$ admissible and makes the value independent of an ambient enlargement; parameterization invariance, this ambient-domain independence, and agreement with the continuous line integral are the well-definedness obligations recorded for The rho-length and the extremal length are well defined ↗. (3) The page's rectangle and round-annulus constants are fixed by the computations stated there, and each cited source is translated into this library convention.