Hoi, ive been breaking my head on this fora few days.. Ive been trying to show that $T:[0,1)\to [0,1)$ given by
$ T(x)= \begin{cases} 3x & \mbox{ if } x\in [0,1/3)\\ & \\ \frac{3x}{2}-\frac{1}{2} & \mbox{ if } x\in [1/3,1) \end{cases} $ is Ergodic.
Let B be T-invariant,i.e. $T^{-1}B=B$. If $\mu(B)>0$, I want to show that $\mu(B)=1$.
Can we show that if $\mu(B)<1$ that $\mu(TB)>\mu(B)$,
is that obvious? So that it is clear that B can not T-invariant?
We know nothing of the set B, which may be little complicated set. Any idea how we can show ergodic property of this transformation. (it is measure preserving).