  It is shown that Bogoliubov quasi-averages select the pure or ergodic states
in the ergodic decomposition of the thermal (Gibbs) state. Our examples include
quantum spin systems and many-body boson systems. As a consequence, we
elucidate the problem of equivalence between Bose-Einstein condensation and the
quasi-average spontaneous symmetry breaking (SSB) discussed for continuous
boson systems. The multi-mode extended van den Berg-Lewis-Pul\'{e} condensation
of type III demonstrates that the only physically reliable quantities are those
that defined by Bogoliubov quasi-averages.
