Given a probability space $$ (\Omega , {\mathcal {A}}, P) $$ , a complete and separable metric space X with the $$ \sigma $$ -algebra $$ {\mathcal {B}} $$ of all its Borel subsets and a $$ {\mathcal {B}} \otimes {\mathcal {A}} $$ -measurable $$ f: X \times \Omega \rightarrow X $$ we consider its iterates $$ f^n$$ defined on $$ X \times \Omega ^{{\mathbb {N}}}$$ by $$f^0(x, \omega ) = x$$ and $$ f^n(x, \omega ) = f\big (f^{n-1}(x, \omega ), \omega _n\big )$$ for $$n \in {\mathbb {N}}$$ and provide a simple criterio…