A finite periodic chain need not converge in total variation

Let $\varphi$ be counting measure. Every nonempty set is reachable from either state at a positive time of at most two steps, so [F2] holds. The uniform law is invariant because each column sum and row sum of the transition matrix equals one.

The partition ${\mathsf{X}}_0=\{0\}$, ${\mathsf{X}}_1=\{1\}$ has positive invariant mass in both classes and transitions alternate deterministically, so it is a cyclic decomposition of period two under [F3].

From $0$, $P^n(0,\cdot )={\delta }_0$ for even $n$ and ${\delta }_1$ for odd $n$. Taking event $\{0\}$ in [F4], the distance to $\pi$ equals $1\mathrm{/}2$ for every $n$. From $1$ the even and odd laws are interchanged, giving the same distance. Hence convergence from either fixed state fails despite irreducibility and invariance; an initial law equal to $\pi$ remains invariant.