Let $(I_{n})$ be a countable infinite disjoint partition of $[0,1)$ into halfopen intervals. Let $f:[0,1)\to [0,1)$ be the piecewise linear expanding map with $f(I_{n})=[0,1)$ for all $n$. I suppose that the Lebesgue measure is ergodic with respect to $f$, as in the case of $2x$ mod $1$. I think this result should be know, but I do not find a reference. Any help is appreciated.
3

4$\begingroup$ Symbolically, it's a countable Bernoulli shift $(X, \mu, \sigma)$, where $\mu([k])=I_k$. Its ergodicity is a well known fact. It follows from it being mixing, for example. For any two cylinders $A$ and $B$, there exists $N\ge1$ such that $\mu(\sigma^n A\cap B)=\mu(A)\mu(B)$ for all $n\ge N$. $\endgroup$ – Nikita Sidorov Mar 14 at 19:39

$\begingroup$ Dear Nikita, what reference would You use in a paper? $\endgroup$ – Jörg Neunhäuserer Mar 14 at 20:02