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2 deleted 31 characters in body

This is not an answer to the question, but this is too long for a comment.

I give what looks like an example where $f$ is not $L^1$ but where $f(T^n\cdot)/n \to 0$ a.e. I will state it in a probabilistic way (sorry about that).

Let $\mu$ be a probability measure on ${1,2,\dots}$. I assume that $m=\int k\mu(dk)$ is finite. I define a new probability measure by $\hat{\mu}(dk)=k/m.\mu(dk)$. Let us consider a Markov chain $((A_n,B_n))$ on $\Omega={(a,b) \Omega=\{(a,b) : a \ge 1, 1 \le b \le aa\} \subset N \times N$ with the following transitions probabilities :

1) From $(a,b)$ with $b>1$ one goes to $(a,b-1)$.

2) From $(a,1)$ one goes to $(c,c)$ where $c$ is chosen according to $\mu$.

Now let $\nu$ be a probability measure on $\Omega$ defined by $\nu({(a,b)})=\mu({a})/m$. \nu(\{(a,b)\})=\mu(\{a\})/m$. The above Markov chain admits$\nu$as a stationary distribution and I consider it under this distribution. Note that$A_n$is distributed according to$\hat{\mu}$. Now set$X_n=g(A_n)$where$g$is in$L^1(\mu)$but not in$L^1(\hat{\mu})$. I think that$(X_n)_n$is a counterexample (consider the Markov Chain seen at the times at wich which it jumps from$(a,\cdot)$to$(c,c)$, then the first coordinates make a sequence of i.i.d.r.v. distributed according to$\mu$). Note : where did my { and } escaped ? 1 This is not an answer to the question, but this is too long for a comment. I give what looks like an example where$f$is not$L^1$but where$f(T^n\cdot)/n \to 0$a.e. I will state it in a probabilistic way (sorry about that). Let$\mu$be a probability measure on${1,2,\dots}$. I assume that$m=\int k\mu(dk)$is finite. I define a new probability measure by$\hat{\mu}(dk)=k/m.\mu(dk)$. Let us consider a Markov chain$((A_n,B_n))$on$\Omega={(a,b) : a \ge 1, 1 \le b \le a} \subset N \times N$with the following transitions probabilities : 1) From$(a,b)$with$b>1$one goes to$(a,b-1)$. 2) From$(a,1)$one goes to$(c,c)$where$c$is chosen according to$\mu$. Now let$\nu$be a probability measure on$\Omega$defined by$\nu({(a,b)})=\mu({a})/m$. The above Markov chain admits$\nu$as a stationary distribution and I consider it under this distribution. Note that$A_n$is distributed according to$\hat{\mu}$. Now set$X_n=g(A_n)$where$g$is in$L^1(\mu)$but not in$L^1(\hat{\mu})$. I think that$(X_n)_n$is a counterexample (consider the Markov Chain seen at the times at wich it jumps from$(a,\cdot)$to$(c,c)$, then the first coordinates make a sequence of i.i.d.r.v. distributed according to$\mu\$).

Note : where did my { and } escaped ?