Strong Law of Large Numbers for Iterates of Some Random-Valued Functions

Abstract Assume $$ (\Omega , {\mathscr {A}}, P) ( Ω , A P ) is a probability space, X compact metric space with the \sigma σ -algebra {B}} B of all its Borel subsets and f: \times \Omega \rightarrow f : X × → \otimes {A}} ⊗ -measurable contractive in mean. We consider sequence iterates f defined on ^{{\mathbb {N}}}$$ N by $$f^0(x, \omega ) = x$$ 0 x ω = f^n(x, f\big (f^{n-1}(x, ), _n\big )$$ n - 1 for $$n \in {\mathbb {N}}$$ ∈ weak limit $$\pi π . show that if $$\psi :X {R}}$$ ψ R continuous, then every x $$\left( \frac{1}{n}\sum _{k=1}^n \psi \big (f^k(x,\cdot )\big )\right) _{n ∑ k · converges almost surely to $$\int _X\psi d\pi ∫ d In fact, we are focusing case where complete separable.