Metric complexity functions measure an amount of instability of trajectories in dynamical systems acting on metric spaces. They reflect an ability of trajectories to diverge by the distance of e during the time interval n. This ability depends on the position of initial points in the phase space, so, there are some distributions of initial points with respect to these features that present themselves in the form of Borel measures. There are two approaches to deal with metric complexities: the one based on the notion of ε-nets (εspanning) and the other one defined through ε-separability. The last one has been studied in [1, 2]. In the present article we concentrate on the former. In particular, we prove that the measure is invariant if the complexity function grows subexponentially in n.
