In Tierney's paper (http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.110.5995&rep=rep1&type=pdf) on page 1712 in the first paragraph there's this proposition that if a chain $P$ is $\pi$-irreducible and it is an invariant distribution for the chain, then the chain must be recurrent.
In the justification, the author showed that if it is not the case, then there exist sets $B_i$ such that $E= \cup B_i$ and the $B_i$ are transient, that is , $P^n (x, B_i) \rightarrow 0$ for all x. Then "since at least one of these sets must have positive $\pi$- probability, this leads to a contradiction".
I have trouble understanding this last statement. Evidently it must be using the irreducibility of the chain to arrive at a contradiction. But how $\pi(B_k) >0 $ and $P^n(x, B_k) \rightarrow 0 $ for some $k$ results in a contradiction escapes me. On the surface of it, it doesn't seem to contradict the definition of $\pi$-irreducibility.
Because $0<\pi(B_k) =\int_E \pi(dx)P^n(x,B_k)$, as $\pi$ is an invariant distribution. If $\lim_n P^n(x,B_k)=0$ for all $x$, then by dominated convergence the integral $\int_E \pi(dx)P^n(x,B_k)$ converges to $0$, which contradicts the positivity of $\pi(B_k)$.