A countably generated phi-irreducible chain has a positive-measure small set

Write $\varphi$ as a countable disjoint union of measurable finite-measure sets $E_j$. Set $c_j=2^{-j}\mathrm{/}(1+\varphi (E_j))$ and $c={\sum }_j{c}_j\varphi (E_j)$. Since $\varphi$ is nonzero and the $E_j$ cover the space, $0<c\le 1$. Then $\bar{\varphi }(A)=c^{-1}{\sum }_j{c}_j\varphi (A\cap E_j)$ is an equivalent probability measure: it has density $c_j\mathrm{/}c>0$ on every $E_j$, and hence exactly the same null sets and positive sets as $\varphi$. Replace $\varphi$ by $\bar{\varphi }$ for the rest of the proof. By [F3], for every $x$ the sum ${\sum }_n{p}_n(x,y)$ is positive for $\varphi$-almost every $y$. Tonelli and countability imply that for ${\varphi }^{\otimes 3}$-almost every $(x,y,z)$, this sum is positive at $(x,y)$ and the corresponding sum at $(y,z)$ is positive. Hence the product of the two sums is positive almost everywhere; its countable expansion shows that some fixed $r,s\ge 1$ satisfy $\int p_r(x,y)p_s(y,z)\varphi (dx)\varphi (dy)\varphi (dz)>0.$ For a sufficiently small $\eta >0$, the set where both factors exceed $\eta$ still has positive product measure.

Apply [F4] to $A=\{(x,y):p_r(x,y)>\eta \}$ and $B=\{(y,z):p_s(y,z)>\eta \}$. Obtain positive-$\varphi$ sets $C_0,D_0$ and $\gamma >0$ such that their sections overlap by at least $\gamma$. The convolution bound [F3] gives $p_{r+s}(u,v)\ge {\eta }^2\gamma$ for every $u\in C_0,v\in D_0$.

Since $C_0$ is $\varphi$-positive, irreducibility gives ${\sum }_{k\ge 1}{P}^k(x,C_0)>0$ for every $x$. Integrating over $D_0$ and using Tonelli, some $k\ge 1$ has ${\int }_{D_0}{P}^k(x,C_0)\varphi (dx)>0$. A positive-level-set subdivision yields $G\subseteq D_0$ with $\varphi (G)>0$ and $\delta >0$ such that $P^k(x,C_0)\ge \delta$ for every $x\in G$.

For every $x\in G$ and $z\in D_0$, the second convolution inequality in [F3] gives $p_{r+s+k}(x,z)\ge {\int }_{C_0}{P}^k(x,dy)p_{r+s}(y,z)\ge \delta {\eta }^2\gamma \mathrm{.}$ Let $\nu (A)=\varphi (A\cap D_0)\mathrm{/}\varphi (D_0)$ and choose $0<ϵ\le \min \{1,\delta {\eta }^2\gamma \varphi (D_0)\}$. Integrating the density over $A\cap D_0$ gives $P^{r+s+k}(x,A)\ge ϵ\nu (A)$ for every $x\in G$. Since $G\subseteq D_0$ and $\varphi (G)>0$, $\nu (G)>0$. Thus $G$ is the required small set.