  A simple proof of the fact that each rank-one infinite measure preserving
(i.m.p.) transformation is subsequence weakly rationally ergodic is found. Some
classes of funny rank-one i.m.p. actions of Abelian groups are shown to be
subsequence weakly rationally ergodic. A constructive definition of finite
funny rank for actions of arbitrary infinite countable groups is given. It is
shown that the ergodic i.m.p. transformations of balanced finite funny rank are
subsequence weakly rationally ergodic. It is shown that the ergodic probability
preserving transformations of exact finite rank, the ergodic Bratteli-Vershik
maps corresponding to the "consequtively ordered" Bratteli diagrams of finite
rank, some their generalizations and the ergodic IETs are partially rigid.
