Dynamical Systems in Probability Theory

Suppose M is the set of all sequences, infinite in both directions x = (..., y_-1, y₀, y₁,...), whose coordinates y _i are points of a fixed measurable space (Y, ??). M possesses a natural σ-algebra ??̃ generated by cylindrical sets, i.e., sets of the form (1)$$ A = \{ x = (...,{y_{{ - 1}}},{y_{0}},{y_{1}},...) \in M:{y_{{{i_{1}}}}} \in {C_{1}},...,{y_{{{i_{r}}}}} \in {C_{r}}\} , $$ where 1 ≤ r < ∞, i₁,..., i _r are integers and C₁,..., C _r ∈ ??. Suppose μ is a normalized measure on ??̃ and ?? is the completion of ??̃ with respect to the measure μ. In probability theory the triple (M, ??, μ) is said to be a discrete time random process and the space (Y, ??) is the state space of this process.