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 $\rho$.