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We generalize a result of J. C. Kelly to the setting of Ahlfors $Q$-regularmetric measure spaces supporting a $1$-Poincar\'e inequality. It is shown thatif $X$ and $Y$ are two Ahlfors $Q$-regular spaces supporting a $1$-Poincar\'einequality and $f:X\to Y$ is a quasiconformal mapping, then the$Q/(Q-1)$-modulus of the collection of measures$\mathcal{H}^{Q-1}\lfloor_{\Sigma E}$ corresponding to any collection of sets$E\subset X$ of finite perimeter is quasi-preserved by $f$. We also show thatfor $Q/(Q-1)$-modulus almost every $\Sigma E$, if the image surface $\Sigmaf(E)$ does not see the singular set of $f$ as a large set, then $f(E)$ is alsoof finite perimeter. Even in the standard Euclidean setting our results aremore general than that of Kelly, and hence are new even in there.

Modulus of families of sets of finite perimeter and quasiconformal maps between metric spaces of globally $Q$-bounded geometry