It's stated in Väisälä's 'Lectures on n-dimensional quasiconformal mappings' (p. 20) that, in the geometric definition of a quasiconformal mapping, that the modulus of a family of curves associated to a ring is unaffected by considering admissible functions $\rho$ which are continuous and not just measurable. No further explanation was given there. I skimmed through some papers by Gehring, Väisälä, etc. from the time period and couldn't find any elaboration on this. Could someone suggest a reference, preferably with proof?

The question could also be interpreted in terms of the regularity of admissible functions for capacities.

A reminder of the definitions: for any disjoint connected continua $E, F \subset \mathbb{R}^n$, we consider the family $\Gamma$ of curves with initial point in $E$ and terminal point in $F$. A measurable function $\rho: \mathbb{R}^n \rightarrow [0, \infty]$ is admissible if $\int_\gamma \rho ds \geq 1$ for all rectifiable $\gamma \in \Gamma$. Then $\text{Mod}_p \Gamma = \inf\{ \int_{\mathbb{R}^n} \rho^p dm\}$, the infimum taken over all admissible $\rho$.


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Gehring shows (for p=n=3) in the first equality of Theorem 1 in his paper

Extremal length definitions for the conformal capacity of rings in space. https://projecteuclid.org/euclid.mmj/1028998672

that the conformal modulus of a ring domain $R$ (defined via measurable functions) can equivalently be defined as the infimum of $\int |Df|^p$ where $f$ varies among $ACL$ functions having boundary values 0 and 1 on the two boundary components of $R$. Moreover he cites on p.138 another paper of himself showing that $ACL$ may be replaced by $C^1$.

It seems quite likely that a corresponding characterization also holds for general n and $p$ (not necessarily equal to $n$), maybe even the same proof goes through, see edit below.

Assuming that these results also hold for general $p$ and $n$, it follows immediately from the argument on p.142 of the same paper that the conformal modulus of $R$ can also be reached as the infimum over continuous admissible functions.

Edit: Actually the whole argument seems to be essentially spelled out in Rickman's book on quasiregular mappings, starting on p.53, section 10.


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