Phi-irreducibility ($\phi (A)>0⟹{\sum }_{n\ge 1}{P}^n(x,A)>0$)

A Markov chain with transition kernel $P$ is $\phi$-irreducible if $\phi$ is a nonzero sigma-finite measure and every measurable set $A$ with $\phi (A)>0$ satisfies ${\sum }_{n\ge 1}{P}^n(x,A)>0$ from every starting state $x$. This is positive reachability, not certain eventual hitting.