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, $\psi_n T=g_1 \wedge n = g_0 \wedge n = \psi_n.$