$\sigma$-algebra of $\theta$-invariant sets in ergodicity theorem for stationary processes Applying Birkhoff's ergodic theorem to a stationary process (a stochastic process with invariant transformation $\theta$, the shift-operator - $\theta(x_1, x_2, x_3,\dotsc) = (x_2, x_3, \dotsc)$) one has a result of the form

$ \frac{X_0 + \dotsb + X_{n-1}}{n} \to \mathbb{E}[ X_0 \mid J_{\theta}]$ a.s.

where the right hand side is the conditional expectation of $X_0$ concerning the sub-$\sigma$-algebra of $\theta$-invariant sets... How do these sets in $J_{\theta}$ look like? (I knew that $\mathbb{P}(A) \in \{0,1\}$ in the ergodic case, but I don't want to demand ergodicity for now).

The $T$-invariant $\sigma$-field is generated by the (generally nonmeasurable) partition into the orbits of $T$. See https://mathoverflow.net/questions/88268/partition-into-the-orbits-of-a-dynamical-system 
This gives a somewhat geometric view of the invariant $\sigma$-field. You should also studied the related notion of ergodic components of $T$.
