$T^2$ is measure preserving but $T$ is not measure preserving

Question

I have had trouble with this exercise. Can anyone help?

Give an example of a probability space $(\Omega, \mathcal{F}, P)$ and a measurable mapping $T : \Omega \rightarrow \Omega$ such that $T^2$ is measure preserving but $T$ is not measure preserving.

There is a hint that I should consider a space $\Omega$ containing only two points.

Best,

Answer 1

Let $\Omega=\{a,b\}$ with $\mathbb P(\{a\})=1/3, \mathbb P(\{b\})=2/3$ and $T(a)=b$, $T(b)=a$. Then $T^2(a)=a$, $T^2(b)=b$ and this is measure-preserving transformation since it is identity.

But $T$ is not measure preserving: say, for $A=\{a\}$, $\mathbb P(TA)\neq \mathbb P(A)$.