 Solution

Suppose first that the system is ergodic and $K_{F}g=g$. For each real $t$, the level setis invariant modulo a null set, so $\omega (E_t)\in \{0,1\}$. The distribution function of $\mathrm{Re}g$ can therefore jump only once, which makes $\mathrm{Re}g$ constant almost everywhere. The same argument applies to $\mathrm{Im}g$.

Conversely, if $E$ is invariant, then $K_F{1}_E=1_E$. If every invariant $L^2$ function is constant, the indicator function $1_E$ is almost everywhere zero or one, and hence $\omega (E)=0$ or $1$. This proves the invariant-function characterization of ergodicity.