At fust we are devoted primarily to part of the progress made in the last 30-yrs with exposition of the well-known theorems from a methodological viewpoint, stress the most satisfac-tory crgodic approach to statistical mechanics in the version of the ensemble theory from a conceptual point of view and suggest a new approach to the quantum ergodie theorem. Secondly we have the aim to study its mathematical aspect in which we re-consider the problem by the methods of the Markov process and the functional analysis in order to secure the foundation of the measure-theoretic ergodic theory and the existence of the invariant measure.