“…We shall also use the same symbol τ to denote the induced set transformation, which takes, for example, sets B ∈ B m into sets τB ∈ B m+1 ; for instance, see [52]. The naming of strong mixing in the above definition is more stringent than what is ordinarily referred to (when using the vocabulary of measure preserving dynamical systems) as strong mixing, namely to that lim n→∞ P(A ∩ τ −n B) = P(A)P(B) for any two measurable sets A, B; see, for instance [52,53] and more recent references [54][55][56][57][58][59][60]. Hence, strong mixing implies ergodicity, whereas the inverse is not always true (see, e.g., Remark 2.6 in page 50 in connection with Proposition 2.8 in page 51 in [40]).…”