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.



Expectation of a linear operator
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.


