 Solution

Let $P$ be the transition kernel of the Markov chain. It is phi-irreducible if there is a nonzero sigma-finite measure $\phi$ such that, for every measurable $A$ and every starting state $x$,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.