Product of ergodic system with rotation of cyclic group of finite order.

Question

Let $(X,\mu,T)$ be an ergodic system and $(G,\nu,R)$ be a system with $G=<g>$, a cyclic group of finite order, say $k\in \mathbb{N}$, $R(g^m)=g^{m+1}$ and $\nu(g^m)=\frac{1}{k}$, for $m\in \{0,1,...,k-1\}$. Isn't then the system $(X\times G, \mu \times \nu, T\times R)$ ergodic as well?

I can see that for any subset of $X\times G$ of the form $A\times B$, where $A$ is a subset of $X$ and $B$ a subset of $G$ we can only have $(T\times S)^{-1}(A\times B)=A\times B\ $ if $\mu\times \nu(A\times B)=0\ \text{or}\ 1$, but is that enough?

Answer 1

Hint: Put $X=G=\mathbb{Z}/2\mathbb{Z}$ with uniform probability; $T=R:x\mapsto x+1$ (modulo $2$).