Definition (μ-irreducible)

A Markov chain is $\mu$-irreducible if for any $B\in\mathcal{B}(\mathbb{X})$ with positive measure $\mu(B)>0$ and any $x\in\mathbb{X}$, $\exists n>0$ s.t. $$P^{n}(x,B)>0$$i.e. $$\mathbb{P}(x_{t+n}\in B\mid x_{t}=x)>0$$ there exists some finite path to any state with any initial state with positive probability.

Definition (Irreducible (474))

A Markov chain $\{X_i\}$ is called irreducible if one can go from any state value in $\mathcal{X}$ to any other state value in $\mathcal{X}$ in a finite number of transitions with positive probability, i.e., $$\forall a,b\in\mathcal{X} \mbox{ and } i\ge1, \ \exists t\ge1\mbox{ s.t. }P(X_{t+i}=b|X_{i}=a)>0$$ or “you can reach any other state from every state (no closed loops)”.

Definition (Irreducible (455))

We say a transition matrix $P$ is irreducible if for any $i,j\in S$, $$i\leftrightarrow j$$That is, $S$ is a single communicating class. We will also say that the Markov chain is irreducible.

Definition (Maximal Irreducible Measure)

A measure $\psi$ is a maximal μ-irreducible measure if $\forall\mu$ s.t. $\{ X_{i} \}_{i=1}^{\infty}$ is μ-irreducible then $\mu \ll \psi$ (i.e. $\mu$ is absolutely continuous w.r.t. $\psi$) or $$\psi \text{ is maximal irreducible mesaure}\iff\mu\ll \psi, \ \forall\mu \ \mu\text{-irredudible}$$