Given a function $f: \Omega \to \mathbb{R}$, where $\Omega$ is the state space of an ergodic Markov chain. Is it true that $\mathbb{E}_t (f- \mathbb{E}_t f)^2$ is nondecreasing in $t$? Here $\mathbb{E}_t f$ denotes $\mathbb{E} P_t f$ 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.