If $f:[0,1]\to [0,1]$ is given by
$$ f(x)= \begin{cases} 2x & \mbox{ if } x\in [0,1/3)\\ & \\ 2x-\frac{2}{3} & \mbox{ if } x\in [1/3,1/2)\\ 2x-\frac{1}{3} & \mbox{ if } x\in [1/2,2/3)\\ & \\ 2x-1 & \mbox{ if } x\in [2/3,1] \end{cases} $$ show that $f$ is Ergodic with respect of Lebesgue measure.
The idea is to try to show that given an invariant set of positive measure, this set will have full measure.
A very common process is to partition the domain, choose a point of density in the invariant set, use the the Lebesgue Density Theorem and the bounded distortion property
\begin{align*} \dfrac{m(f^k(E_1))}{m(f^k(E_2))}=\dfrac{m(E_1)}{m(E_2)} \end{align*}.
I have several questions. In this case, how should I partition the set? Should I use the iterates of f to form the partition in the domain? Or do not you? What would her iterations look like? What are the conditions for using the motto of limited distortion?