Suppose we have two measurable dynamical systems $(X_1,\mathcal{B}_1,\mu_1,T_1)$ and $(X_2,\mathcal{B}_1,\mu_2,T_2)$, with $\mu_i(X_i)=1,\ i=1,2$. Suppose they are measure-theoretically isomorphic (so there is an almost-sure one-to-one map $\theta:X_1\to X_2$ such that $\theta \circ T_1 = T_2 \circ \theta$ and $\mu_1(T^{-1}(A)) = \mu_2(A)$).
I have a question that is probably very simple but I would like to check myself.
Is true that
- Ergodic decomposition of $\mu_1$ is preserved (ergodic components are mapped to ergodic components with the same "weights")?
- Generic points (those for which ergodic averages converge to the $\mu_1$-space average) of $\mu_1$ are mapped to generic points of $\mu_2$?