Extension of the concept of invariance and statistically weakly invariant sets of controllable systems

We continue the study of statistically invariant and statistically weakly invariant sets with respect to controllable systems and differential inclusions launched by Prof. E. L. Tonkov. We examine properties of such statistical characteristics as the lower $\operatorname{freq}_*(\varphi)$ and upper $\operatorname{freq}^*(\varphi)$ relative frequencies of hitting a solution $\varphi(t)$ of a differential inclusion in a prescribed set. We obtain estimates and conditions of the coincidence of these characteristics for functions whose difference tends to zero at infinity. We also present conditions of statistically weak invariance of a given set of a relatively controllable system.