Suppose $(X,\mu)$ is a probability space, and $T\_n, n \in \mathbb N$, is a sequence of *periodic* measure preserving transformations. For $x \in X$ and $f : X \to \mathbb R$, let $\mathrm{avg}\_{f,n}(x)$ be the average value of the finite set $\{ f(T\_n^0(x)), f(T\_n^1(x)), f(T\_n^2(x)),\dots\}$ (finite because of periodicity).


I want to know, is there a general “ergodic” theorem that tells us under certain conditions on the sequence $T\_n$, $\lim\_{n \to \infty} \mathrm{avg}\_{f,n}(x) =\int f d\mu$ for $\mu$-almost all $x$, when $f$ is integrable?


Note that typically none of the $T\_n$ are ergodic, and they might have a non-ergodic limit. For example, $T\_n$ could be translation by $1/n$ on the unit interval. But I think the equation holds in this case.



Ergodic theorem on limit of periodic transformations?
In fact, for the example you give, there is a failure of pointwise convergence. That was established in a short 1964 paper of [Rudin](https://www.ams.org/journals/proc/1964-015-02/S0002-9939-1964-0159918-8/home.html) in Proc. Amer. Math. Soc. A more detailed look at this example appears in "The strong sweeping out property for lacunary sequences, Riemann sums, convolution powers, and related matters" by Akcoglu, Bellow, Jones, Losert, Reinhold-Larsson and Wierdl (ETDS 1996).


The general conditions to have this kind of limit are:


1. a weak $L^1$ maximal theorem;
2. existence of a dense set of functions $f$ for which convergence occurs.


In your example, you have condition 2. but not condition 1.
For more information, have a look at my answer to [this question](https://mathoverflow.net/questions/390544/almost-surely-convergence-of-translations-of-a-measurable-function/390654#390654), as well as some of the references that appear as comments in that question.


