Let $(X,\sigma,\mu)$ be a measure space, $\mu(X)=1$. End let $T:X\to X$ be a measurable map such that 
$$D\in\sigma\Longrightarrow \mu(T^{-1}D)=\mu(D).$$ Let $f:X\to\mathbb{R}_+$ be an integrable function. Construct a measurable function as follows
$$\psi_n:=\Big(\limsup_{k\to\infty}\frac{1}{k}\sum_{i=0}^{k-1}fT^i\Big)\wedge n.$$
The paper 
[Easy and nearly simultaneous proofs of the Ergodic Theorem and Maximal Ergodic Theorem][1]
claims (before the ergodic theorem!) that for all $n\in\mathbb{N}$ the functions $\psi_n$ are  invariant:
$$\psi_n T=\psi_n,\quad \mbox{a. e.}$$
Please could you push me in the direction of the proof. It must be easy but I am stuck.
 


  [1]: https://projecteuclid.org/ebooks/institute-of-mathematical-statistics-lecture-notes-monograph-series/Dynamics-amp-Stochastics/chapter/Easy-and-nearly-simultaneous-proofs-of-the-Ergodic-Theorem-and/10.1214/074921706000000266