On the closedness of ergodic measures in a characteristic class

We endow the set of all invariant measures of a topological dynamical system with a metric $\bar{\rho}$, which induces a topology stronger than the the weak$^*$-topology. Then, we study the closedness of ergodic measures within a characteristic class under this metric. Specifically, we show that if a sequence of generic points associated with ergodic measures from a fixed characteristic class converges in the Besicovitch pseudometric, then the limit point is generic for an ergodic measure in the same class. This implies that the set of ergodic measures belonging to a fixed characteristic class is closed in $\bar{\rho}$ (by a result of Babel, Can, Kwietniak, and Oprocha in [1]).