Let  $G$  be  a  compact topological group with normalized  Haar measure $\mu$. An  element $g\in G$ is  an ergodic element if the mapping $L_g:G \to G $  with $x\mapsto gx$ is  an ergodic map.The set of ergodic elements of a group $G$ is denoted by $E(G)$

>Is there  a  compact topological group or a compact Lie group for  which the set of  all ergodic elements have intermediate  measuare  namely  we have $0<\mu(E(G)<1$?