Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 16 / 5 / i / Solution

Here is an alternative route for the ergodicity step using the Birkhoff ergodic theorem. After invariance has been proved, the theorem identifies the almost-everywhere limit of $A_n\phi$ with ${\mathbb{E}}_{\mu }[\phi \mid \mathcal{I}]$. The hypothesis therefore givesfor every continuous function $\phi$. For an invariant Borel set $B$, approximate $h=1_B$ in $L^1$ by such a $\phi$. Since $h$ is measurable with respect to the invariant sigma-algebra, its conditional expectation is $h$ itself. The L1 contraction of conditional expectation givesArbitrarily accurate continuous approximation forces $h=\mu (B)$ almost everywhere. Thus this use of the theorem reaches the same zero-or-one criterion as the direct averaging proof above, without assuming in advance that $\mu$ is ergodic.