= Solution
Suppose first that the system is <ergodic measure-preserving transformation>[ergodic] and $K_Fg=g$. For each real $t$, the level set
$$
E_t=\{x:\operatorname{Re}g(x)>t\}
$$
is invariant modulo a null set, so $\omega(E_t)\in\{0,1\}$. The distribution function of $\operatorname{Re}g$ can therefore jump only once, which makes $\operatorname{Re}g$ constant almost everywhere. The same argument applies to $\operatorname{Im}g$.
Conversely, if $E$ is invariant, then $K_F\mathbf1_E=\mathbf1_E$. If every invariant $L^2$ function is constant, the <indicator function> $\mathbf1_E$ is almost everywhere zero or one, and hence $\omega(E)=0$ or $1$. This proves the <invariant-function characterization of ergodicity>.
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