= Solution
Let $P$ be the <Markov kernel> and $\mu$ its invariant <probability measure>. Using the convention
$$
\|\nu-\mu\|_{\mathrm{TV}}=\sup_A|\nu(A)-\mu(A)|,
$$
<geometric ergodicity> means that there are $r\in(0,1)$ and a finite measurable state-dependent prefactor $M(x)$ such that
$$
\boxed{\|P^k(x,\cdot)-\mu\|_{\mathrm{TV}}\leq M(x)r^k\qquad(k\geq0).}
$$
Here <total variation distance> is used, and the rate $r$ does not depend on $x$. The prefactor need not be uniformly bounded; that stronger property is <uniform geometric ergodicity>. For a deterministic starting state, this estimate must hold at that state. If the definition is made only $\mu$-almost everywhere, the subsequent assertions concern those starting states where the prefactor is finite. The <reversible Markov chain> assumption means $\mu(dx)P(x,dy)=\mu(dy)P(y,dx)$.
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