Could anyone help me with this? Since I didn't get any answer from math and stat stackexchange so I am writing here.

We define $T: C[0,1]\to C[0,1]\ni T(f(x))= \sum_{k=1}^{m} p_k (f\circ f_k)(x):=\mathbb E( f(X_{n+1}|X_n=x)$ for a system $X_{n+1}=f_{\omega_n}(X_n), n=0,1,2\dots,$ $\omega_n$ 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.