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We prove the existence and uniqueness of the canonical conformally covariantvolume measure on the carpet/gasket of a conformal loop ensemble (CLE$_\kappa$,$\kappa \in (8/3,8)$) which respects the Markov property for CLE. The startingpoint for the construction is the existence of the canonical measure on CLE inthe context of Liouville quantum gravity (LQG) previously constructed by thefirst author with Sheffield and Werner. As a warm-up, we construct the naturalparameterization of SLE$_\kappa$ for $\kappa \in (4,8)$ using LQG which servesto complement earlier work of Benoist on the case $\kappa \in (0,4)$.

Existence and uniqueness of the conformally covariant volume measure on conformal loop ensembles