On ergodic theory

Question

Suppose there exist an action of group $G$ on $L^{\infty}(X,\mu)$ via measure preserving transformation ( the left translation Koopmans action). $\mu$ is probability measure. Suppose the action is ergodic on the subalgebra of simple functions. Can we say the action is ergodic on $L^{\infty}(X,\mu)$?

Answer 1

Of course. Say $X = X_1\sqcup X_2$ with $X_1,X_2$ both $G$-invariant and $\mu(X_1), \mu(X_2) \not \in \{0,1\}$. Then let $f = 1$ on $X_1$ and $0$ on $X_2$. This is a $G$-invariant simple function.