Why should ${\cal E}_\theta$ be trivial?. I dont see how Kolmogorov's 0-1 law says that in this case we should take the 0 option.
This is only mention of ${\cal E}_\theta$ in my notes I can find.
2026-03-27 00:58:35.1774573115
Using Kolmogorov's 0-1 law in proof of shift map being ergodic
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A sigma-algebra is trivial when it contains only sets of measure $0$ or $1$. There is no "0 option" here, that "we should take".