Given a function $f: \Omega \to \R$, where $\Omega$ is the state space of an ergodic Markov chain. Is it true that $\E_t (f- \E_t f)^2$ is nondecreasing in $t$? Here $\E_t f$ denotes $\E P_t f$ where $P_t$ is the Markov transition kernel from time $0$ to time $t$. I feel it's wrong because otherwise it would be too useful, but I can't find a counterexample.