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I am taking my first steps in Ergodic theory and was wondering about a thing:

Given the measure $\mu: R \rightarrow R$ s.t.

$$ \mu(B) = \int_{B} \frac{1}{\sqrt {2\pi }} e^{\frac{x^{2}}{2}} \, dx $$

Does there exist a $\mu$ preserving non trivial map $T: R \rightarrow R$ such that $T$ is ergodic, i.e., for every $B$ in the Borellians s.t. $T^{-1}(B) = B$ we have $\mu(B) \in \{0,1 \} $ ?

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Let $F$ be the cumulative distribution function of the probability measure $\mu$, so $F: \mathbb R \to (0,1)$ is $1-1$ onto and $\mu(A) = m(F(A))$ for any Borel set $A$, where $m$ is Lebesgue measure. Take any ergodic Lebesgue measure preserving map $S$ on $(0,1)$, and $T = F^{-1} \circ S \circ F$ is $\mu$-preserving and ergodic on $\mathbb R$.

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  • $\begingroup$ Thank you very much! could you expand on why $T$ is $\mu$-preserving? $\endgroup$ – Monolite Nov 21 '17 at 22:50
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    $\begingroup$ $\mu(T^{-1}(A)) = \mu(F^{-1}(S^{-1}(F(A)))) = m(S^{-1}(F(A))) = m(F(A)) =\mu(A)$. $\endgroup$ – Robert Israel Nov 22 '17 at 2:41

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