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 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.

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 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.

It is a question for the moderators. I translated into Russian the article cited above and while I was doing it I simplified the proof and even generalized some assertions a bit. If it is interesting I can translate this new version of the article back in English and leave a reference here.

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 claims that for all $n\in\mathbb{N}$ the functions $\psi_n$ are invariant: $$\psi T=\psi,\quad \mbox{a. e.}$$ Please could you push me in the direction of the proof. It must be easy but I am stuck.

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 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.