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Let $\mathbb{P}_0$ be a probability measure on $\mathbb{R}$. Let $\Omega = \mathbb{R}^{\mathbb{Z}^d}$ and $\mathbb{P} = (\mathbb{P}_0)^{\otimes \mathbb{Z}^d}$ so the the canonical process $X:\Omega \rightarrow \Omega$, defined by $X(\omega) =\omega$, can be regarded as iid random variables on the lattice $\mathbb{Z}^d$. The system is translationally-invariant (ergodic), in the sense that if we define $\tau_n :\Omega \rightarrow \Omega$ by $X_m (\tau_n (\omega)) =X_{m-n} (\omega)$, then $\tau_n$ preserves probability $\mathbb{P}$ and all invariant sets (sets $A$ such that $\tau_n^{-1}(A) =A$ for all $n\in \mathbb{Z}^d$) either have probability 0 or 1. Therefore, it seems that we should be able to have a variation of Birkhoff's ergodic theorem, in the sense that, almost surely, we have $$ \frac{1}{(2L+1)^d}\sum_{n\in \Lambda_L} f(X_n)\rightarrow\mathbb{E}f(X_0) $$ where $\Lambda_L =\{-L,-L+1,...,L-1\}^d$.

However, I have only seen Birkhoff's theorem on $\mathbb{N}$ with a single ergodic $\tau: \Omega \rightarrow \Omega$. I'm not sure whether what I'm stating here is true, and if so, the details of its proof. Any references or sketches of the proof would be appreciated.

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The averaging depends often on the particular choice of $\Lambda_L$ but by now results in the direction that you mention should be considered well known.

See for example the paper "Pointwise theorems for amenable groups" by Elon Lindenstrauss in Inventiones Mathematicae 146 (2001), 259-295.

https://link.springer.com/article/10.1007/s002220100162

For a more down to earth reference and specifically for $\mathbb Z^d$ only, have a look at Keller's book Equilibrium States in Ergodic Theory, Cambridge University Press, 1998.

https://www.cambridge.org/core/books/equilibrium-states-in-ergodic-theory/4D3BF3EC4BD0C957FE1F076BD1379710

As always, one should be careful for $d>1$ with the lack of uniqueness of equilibrium measures.

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  • $\begingroup$ Sure, see my answer. $\endgroup$
    – John B
    Commented May 24, 2020 at 23:42

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