= Solution
Let $P$ be the <transition kernel> of the <Markov chain>. It is <phi-irreducible> if there is a nonzero <sigma-finite measure> $\varphi$ such that, for every measurable $A$ and every starting state $x$,
$$
\boxed{\varphi(A)>0\quad\Longrightarrow\quad\sum_{n=1}^\infty P^n(x,A)>0.}
$$
Equivalently, there is positive probability of eventually reaching every set of positive reference measure. For a finite state space with counting measure, this becomes ordinary <irreducible Markov chain> accessibility. On a continuous space, requiring access to each individual point would usually be inappropriate; positive-measure sets play that role. Irreducibility requires positive probability, rather than probability one, and does not itself assert recurrence or a finite mean return time.
Back to article page
