Mean Distality and Tightness

A relationship between the entropy invariant and a certain property of topological dynamical systems with a finite invariant measure $\mu$ is studied. This property means that, after removing a $\mu$-null set, there are no distinct mean proximal points in the system (a pair $x,y$ is mean proximal with respect to a homeomorphism $T$ of a compact metric space with a metric $d$ if $\varlimsup\frac1{2n+1}\sum^n_{-n} d(T^ix, T^iy) = 0$).