The rho-length and the extremal length are well defined

Fix a rectifiable $\gamma :[a,b]\to \mathrm{\Omega }$, let $s=s_{\gamma }$ and $L=L(\gamma )$, and let ${\nu }_{\gamma }$ be the measure of [F2]. Define a finite Borel measure $\eta$ on $\mathbb{R}$ by $\eta (B):={\nu }_{\gamma }(\{t\in [a,b]:s(t)\in B\})$; it is a measure because inverse images preserve Borel sets, disjointness, and countable unions. For $t\in [0,L]$, let $u_t$ be the largest point in the nonempty compact set $\{u\in [a,b]:s(u)\le t\}$. Continuity and surjectivity of $s$ give $s(u_t)=t$ and $\{s\le t\}=[a,u_t]$. Therefore, for $0\le r<q\le L$, $s^{-1}((r,q])\cap [a,b]=(u_r,u_q]$ and [F2] gives $\eta ((r,q])=S(u_q)-S(u_r)=q-r$. The measure $\eta$ is supported on $[0,L]$. Each level set $s^{-1}(\{t\})$ is a closed interval or a singleton; on a nondegenerate level interval $[v_t,u_t]$, [F3] gives ${\nu }_{\gamma }([v_t,u_t])=S(u_t)-S(v_t^{-})=t-t=0$, and on a singleton [F3] gives zero mass as well. Thus $\eta$ has no endpoint atoms. Clipping each half-open interval to $[0,L]$ shows that its mass agrees with Lebesgue measure restricted to $[0,L]$; [F6] gives $\eta =\lambda {\mathrm{\mid }}_{[0,L]}$.

Fix $a\le u\le v\le b$. By [F1], the arc-length function of $\gamma {\mathrm{\mid }}_{[u,v]}$ is $s_{\gamma }(t)-s_{\gamma }(u)$ on $[u,v]$, extended constantly outside that interval. Its associated Lebesgue-Stieltjes measure agrees with ${\nu }_{\gamma }$ restricted to $[u,v]$: both are finite Borel measures supported there and have the same values on every half-open subinterval by [F2] and the interval formulas [F3], so [F6] identifies them. The definition of ${\mathrm{\ell }}_{\rho }$ then gives the displayed restriction formula in (ii). For finitely many subintervals with disjoint interiors, their common endpoints have zero ${\nu }_{\gamma }$-measure by [F3], and additivity of the nonnegative integral [F8] bounds the sum of their path lengths by the integral over $[a,b]$. Taking increasing finite partial sums gives the same bound for a countable family by [F7], proving (ii).

If $\rho$ is finite-valued and continuous, then $g=\rho \circ \gamma$ is a continuous real function. By [F3], ${\nu }_{\gamma }$ has no atoms, so its mass at the endpoints is zero. The continuous-integrand comparison in [F10] identifies ${\mathrm{\ell }}_{\rho }(\gamma )={\int }_{(a,b]}gd{\nu }_{\gamma }$ with the Riemann-Stieltjes absolute line integral, proving the first assertion of (iii). When $\rho =1$, every Riemann-Stieltjes sum against $s_{\gamma }$ telescopes to $s_{\gamma }(b)-s_{\gamma }(a)=L(\gamma )$, proving the final assertion.

If $\rho \le \sigma$, [F8] gives ${\mathrm{\ell }}_{\rho }(\gamma )\le {\mathrm{\ell }}_{\sigma }(\gamma )$ for each path, and taking infima over $\mathrm{\Gamma }$ preserves the inequality. For pairwise disjoint Borel $E_1,\dots ,E_k$ with $\rho =0$ off their union, ${\rho }^2={\sum }_{j=1}^k{\rho }^2{\mathbf{1}}_{E_j}$ pointwise, so finite additivity and the definition of the restricted integral in [F8] give the area sum in (iv).

Apply [F9] to ${\rho }^2$ to obtain $A(\rho )=0$ iff ${\rho }^2=0$ almost everywhere, which is equivalent to $\rho =0$ almost everywhere. For $c>0$, [F8] gives ${\mathrm{\ell }}_{c\rho }(\gamma )=c{\mathrm{\ell }}_{\rho }(\gamma )$ for each path and hence ${\mathrm{\ell }}_{c\rho }(\mathrm{\Gamma })=c{\mathrm{\ell }}_{\rho }(\mathrm{\Gamma })$; it also gives $A(c\rho )=c^{2}A(\rho )$. For $0<A(\rho )<+\mathrm{\infty }$ and real $c>0$, the area $A(c\rho )=c^{2}A(\rho )$ remains finite and positive. If the path-family length is finite, cancellation of $c^2$ proves quotient invariance, including length zero. If the length is $+\mathrm{\infty }$, both quotients are $+\mathrm{\infty }$ because their denominators are finite and positive. This proves (vi) on the full admissible-density domain without an undefined $+\mathrm{\infty }\mathrm{/}+\mathrm{\infty }$ expression.

For an indicator $g={\mathbf{1}}_B$, the definition of $\eta$ gives ${\int }_{[a,b]}g(s(t))d{\nu }_{\gamma }(t)=\eta (B)={\int }_{[0,L]}gds$. Finite sums give the same identity for nonnegative simple $g$, and [F7] extends it to every nonnegative Borel $g$ by increasing simple approximation and monotone convergence. Taking $g=\rho \circ \tilde{\gamma }$ and using $\gamma =\tilde{\gamma }\circ s$ from [F4] yields ${\mathrm{\ell }}_{\rho }(\gamma )={\int }_0^L\rho (\tilde{\gamma }(s))ds$.

Extending $\rho$ by zero from $\mathrm{\Omega }$ to ${\mathrm{\Omega }}^{\mathrm{\prime }}$ preserves each path integral for paths in $\mathrm{\Omega }$ and preserves area, because the new integrand is zero off $\mathrm{\Omega }$. Extension therefore gives ${\lambda }_{{\mathrm{\Omega }}^{\mathrm{\prime }}}(\mathrm{\Gamma })\ge {\lambda }_{\mathrm{\Omega }}(\mathrm{\Gamma })$. Conversely, take any ${\rho }^{\mathrm{\prime }}:{\mathrm{\Omega }}^{\mathrm{\prime }}\to [0,+\mathrm{\infty }]$ with $0<A_{{\mathrm{\Omega }}^{\mathrm{\prime }}}({\rho }^{\mathrm{\prime }})<\mathrm{\infty }$, and put $\rho ={\rho }^{\mathrm{\prime }}{\mathrm{\mid }}_{\mathrm{\Omega }}$. Its path-length infimum on $\mathrm{\Gamma }$ is unchanged and $A_{\mathrm{\Omega }}(\rho )\le A_{{\mathrm{\Omega }}^{\mathrm{\prime }}}({\rho }^{\mathrm{\prime }})$. If $A_{\mathrm{\Omega }}(\rho )>0$, this restriction is admissible and its quotient is at least the quotient from ${\mathrm{\Omega }}^{\mathrm{\prime }}$. If $A_{\mathrm{\Omega }}(\rho )=0$, [F9] gives $\rho =0$ almost everywhere. When ${\mathrm{\ell }}_{\rho }(\mathrm{\Gamma })=0$, the quotient from ${\mathrm{\Omega }}^{\mathrm{\prime }}$ is zero. When ${\mathrm{\ell }}_{\rho }(\mathrm{\Gamma })>0$, choose the rectangle $Q$ of [F11] and, for $\epsilon >0$, set ${\rho }_{\epsilon }=\rho +\epsilon {\mathbf{1}}_Q$ on $\mathrm{\Omega }$. Then $A_{\mathrm{\Omega }}({\rho }_{\epsilon })={\epsilon }^2\mathrm{\mid }Q\mathrm{\mid }$ and ${\mathrm{\ell }}_{{\rho }_{\epsilon }}(\mathrm{\Gamma })\ge {\mathrm{\ell }}_{\rho }(\mathrm{\Gamma })$ by [F8], so the quotients on $\mathrm{\Omega }$ are unbounded as $\epsilon \downarrow 0$. Thus every admissible quotient on ${\mathrm{\Omega }}^{\mathrm{\prime }}$ is at most ${\lambda }_{\mathrm{\Omega }}(\mathrm{\Gamma })$, proving equality of extremal lengths; their reciprocals agree as well.

Let $h:[a^{\mathrm{\prime }},b^{\mathrm{\prime }}]\to [a,b]$ be continuous, nondecreasing and onto. By [F5], the total lengths of $\gamma \circ h$ and $\gamma$ agree, and applying [F5] to each restriction gives $s_{\gamma \circ h}=s_{\gamma }\circ h$. Thus $\gamma \circ h=\tilde{\gamma }\circ s_{\gamma \circ h}$ with the same unique arc-length parametrization $\tilde{\gamma }$ as in [F4]. The formula of step 2.1 applied to both paths proves ${\mathrm{\ell }}_{\rho }(\gamma \circ h)={\mathrm{\ell }}_{\rho }(\gamma )$, establishing (i).

Steps 1.1, 2.1, and 3.1 prove parameterization independence and the arc-length formula in (i); step 1.2 proves (ii), step 1.3 proves (iii), step 1.4 proves (iv), step 2.2 proves (v), and step 1.5 proves (vi).