TITLE: Ergodicity of tent map
QUESTION [7 upvotes]: The dynamical system $T:[0,1]\to [0,1]$ defined by
$$T(x) = \begin{cases}
       2x, & \text{for } 0\leq x\leq \frac{1}{2}\\
        2-2x, & \text{for }  \frac{1}{2}\leq x\leq 1
        \end{cases}$$
is called the tent map. Prove that $T$ is ergodic with respect to Lebesgue measure.
My work:
Measure preserving map is ergodic if for every T-invariant measurable set is $m(A)=1$ or $m(A)=0$ (by definition). 
Let's prove that $T$ preserves Lebesgue measure. It is enough to prove for intervals $[a,b]$ that $m(T^{-1}[a,b])=m([a,b])$ holds. Since $T^{-1}[a,b]=[\frac{a}{2},\frac{b}{2}]\cup [\frac{2-b}{2},\frac{2-a}{2}]$, we have $m(T^{-1}[a,b])=m([\frac{a}{2},\frac{b}{2}]\cup [\frac{2-b}{2},\frac{2-a}{2}])=m([\frac{a}{2},\frac{b}{2}])+m([\frac{2-b}{2},\frac{2-a}{2}])=b-a=m([a,b])$. Therefore, $T$ is measure-preserving. 
Now, I'm stuck and don't know how to prove that map is ergodic. Any help is welcome. Thanks in advance.

REPLY [3 votes]: There is a pretty standard way to prove ergodicity of certain class of maps: use the Lebesgue Density Theorem and the following bounded distortion property
\begin{align*}
\dfrac{m(T^k(E_1))}{m(T^k(E_2))}=\dfrac{m(E_1)}{m(E_2)}
\end{align*}
for every pair of measurable subsets $E_1,E_2$ of any interval $I$ where $T^k$ is a bijection. If you need further details, do not hesitate to ask.