I have been reading stuff about ergodic theory, and I have encountered two versions of the involved "invariant sigma field". let the underlying probability space be $(\Omega,\mathcal{F},P)$, and let's consider a measure preserving transformation $T:\Omega \rightarrow \Omega$). then the two definitions of the $T$-invariant sets are:
1) $\mathcal{I}_1:=\{A \in \mathcal{F} \mid T^{-1}A=A\}$;
2) $\mathcal{I}_2:=\{A \in \mathcal{F} \mid P(T^{-1}A \Delta A) = 0\}$;
obviously, $\mathcal{I}_1\subseteq\mathcal{I}_2$. but I don't think that $\mathcal{I}_2\subseteq\mathcal{I}_1$ holds as well. 2hat is the relationship between the two (is the larger one the completion of the smaller one?)?
But here is the prime question: which one is the right "conditioning" sigma field for the limit in Birkhoff's ergodic theorem? Are the conditional expectations versions of each other? If yes, how would you show that?
Many thanks for any help!
Let $\mathcal N$ denote the subsets $N$ of $\Omega$ for $N\in \mathcal F$ and $\mu(N)=0$.
We actually have (thanks Harry for fixing that) $\mathcal I_2=\sigma(\mathcal I_1,\mathcal N)$.
Indeed, if $B\in\mathcal I_2$, then $\mu(B\Delta T^{-1}B)=0$. We define $C:=\bigcap_{N\geqslant 0}\bigcup_{n\geqslant N}T^{-n}B$. Then $\mu(C\Delta B)=0 $ and $C\in\mathcal I_1$. We thus take $N:=C\Delta B$ and observe that $$B = C \setminus (C \cap N) \cup ( B \cap N) \in \sigma(\{\mathcal{I}_1, \mathcal{N}\}).$$
Let us prove that $\mu(C\Delta B)=0 $. From the fact that $\mu(B\Delta T^{-n}B)=0$ for each $n$, and defining $C_N:=\bigcup_{n\geqslant N}T^{-n}B$, we get $\mu(B\Delta C_N)=0$ for each $N$. Now notice that $C_N\downarrow C$.