Assume (Ω,A,P)\documentclass[12pt]{minimal}
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\begin{document}$$ (\Omega , {\mathscr {A}}, P) $$\end{document} is a probability space, X is a compact metric space with the σ\documentclass[12pt]{minimal}
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\begin{document}$$ \sigma $$\end{document}-algebra B\documentclass[12pt]{minimal}
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\begin{document}$$ {\mathscr {B}} $$\end{document} of all its Borel subsets and f:X×Ω→X\documentclass[12pt]{minimal}
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\begin{document}$$ f: X \times \Omega \rightarrow X $$\end{document} is B⊗A\documentclass[12pt]{minimal}
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\begin{document}$$ {\mathscr {B}} \otimes {\mathscr {A}} $$\end{document}-measurable and contractive in mean. We consider the sequence of iterates of f defined on X×ΩN\documentclass[12pt]{minimal}
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\begin{document}$$ X \times \Omega ^{{\mathbb {N}}}$$\end{document} by f0(x,ω)=x\documentclass[12pt]{minimal}
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\begin{document}$$f^0(x, \omega ) = x$$\end{document} and fn(x,ω)=f(fn-1(x,ω),ωn)\documentclass[12pt]{minimal}
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\begin{document}$$ f^n(x, \omega ) = f\big (f^{n-1}(x, \omega ), \omega _n\big )$$\end{document} for n∈N\documentclass[12pt]{minimal}
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\begin{document}$$n \in {\mathbb {N}}$$\end{document}, and its weak limit π\documentclass[12pt]{minimal}
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\begin{document}$$\pi $$\end{document}. We show that if ψ:X→R\documentclass[12pt]{minimal}
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\begin{document}$$\psi :X \rightarrow {\mathbb {R}}$$\end{document} is continuous, then for every x∈X\documentclass[12pt]{minimal}
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\begin{document}$$ x \in X $$\end{document} the sequence 1n∑k=1nψ(fk(x,·))n∈N\documentclass[12pt]{minimal}
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\begin{document}$$\left( \frac{1}{n}\sum _{k=1}^n \psi \big (f^k(x,\cdot )\big )\right) _{n \in {\mathbb {N}}}$$\end{document} converges almost surely to ∫Xψdπ\documentclass[12pt]{minimal}
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\begin{document}$$\int _X\psi d\pi $$\end{document}. In fact, we are focusing on the case where the metric space is complete and separable.
