# Expectation of a linear operator

We define $$T: C[0,1]\to C[0,1]\ni T(f(x))= \sum\limits_{k=1}^{m} p_k (f\circ f_k)(x):=\mathbb E( f(X_{n+1}|X_n=x)$$ for a system $$X_{n+1}=f_{\omega_n}(X_n), n=0,1,2\dots.$$ and $$\omega_n$$ are i.i.d discrete r.v over $$\{1,2,\dots,m\}$$, $$p_k=\text{ Prob} (\omega_i=k)$$, $$f_k$$ are bounded Lipschitz.

Could anyone explain to me how come $$| T^n(f(x))-T^n(f(y))|=| \mathbb E(f(X_n(x))-f(X_n(y)))|$$?

Thanks for the help.

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By the tower law, $$\mathbb{E} [f(X_n) | X_0=x] = \mathbb{E} [\mathbb{E} \{f(X_n) | X_{n-1}\} | X_0=x].$$ Observe that $$\mathbb{E} \{f(X_n) | X_{n-1}\}=Tf(X_{n-1})$$ (because $$(X_n)_{n\in \mathbb{N}}$$ forms a time-homogeneous Markov chain) . By induction it follows that $$E[f(X_n) | X_0=x]=T^n f(x)$$ and your result follows.