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](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)
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.




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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.



near ergodic theory question
I believe you are referring to the function $\lambda$ defined in the first paragraph of page $250$ which carries an extra $-1/n$ (this is irrelevant for the invariance). If I am not missing anything, you have the following.


Define $g\_{\ell}\overset{\Delta}= \limsup\_{k\rightarrow \infty} \frac{1}{k} \sum\_{i=0}^{k-1} fT^{i+\ell}.$


Then, $g\_{\ell}=g\_0$ for all $\ell\in\mathbb{N}$ since


$$g\_{0} = \limsup\_{k\rightarrow \infty} \frac{1}{k} \sum\_{i=0}^{k-1} fT^{i}= \underbrace{\limsup\_{k\rightarrow \infty} \frac{1}{k} \sum\_{i=0}^{\ell-1} fT^{i}}\_{=0}+\underbrace{\limsup\_{k\rightarrow \infty} \frac{k-\ell}{k} \frac{1}{k-\ell}\sum\_{i=\ell}^{k-1} fT^{i}}\_{=g\_{\ell}}.$$


Thus, $\psi\_n T=g\_1 \wedge n = g\_0 \wedge n = \psi\_n.$


