$\pi(S_t)$ is a martingale

Let $P$ be a Markov chain on a finite state space $V$ with strictly positive stationary distribution $\pi$, and write $Q(S,y)=\sum_{x\in S}\pi(x)P(x,y)$ for the stationary flow from a set $S$ into a state $y$. The evolving-set process is the Markov chain on subsets of $V$ that, from $S$, draws $u$ uniform on $(0,1]$ and passes to the superlevel set $\{y:Q(S,y)/\pi(y)\ge u\}$; its transition probability from $S$ to $T$ is the length of the interval of thresholds realizing $T$. A process adapted to a chain is a martingale when its one-step conditional expectation is neutral (the pointwise finite-sum identity of this mission's definitions).

The theorem (Lemma 17.13 of Levin–Peres–Wilmer) asserts that the stationary mass of the evolving set,

is a martingale for the evolving-set process: for every set $S$, $\;\sum_TK(S,T)\,\pi(T)=\pi(S)$, where $K$ is the process's transition matrix.

The mass gained when the threshold $u$ is small exactly balances the mass lost when it is large — stationarity of $\pi$ in disguise. This martingale is the engine of the whole chapter: the goal theorem controls the square root $\sqrt{\pi(S_t)}$ as a strict supermartingale whose decay rate is the bottleneck constant, and optional stopping converts that decay into mixing bounds.