distribution measure of trajectory space in finite state irreducible Markov Chain

Question

Consider a finite state irreducible Markov Chain $(Z_n)$ on the trajectory space $(\Omega, \mathcal{A}, \mathbb{P})$ induced by $(X,P)$ where $X$ is the state space and $P$ a stochastic matrix. For any event $A\in\mathcal{A}$, we have $\mathbb{P}(A)=\sum_{j\in X}\pi(j)\mathbb{P}(A|Z_0=j)$ where $\pi$ is an initial distribution. My question is:

1. If $A$ is a shift-invariant event, do we have $\mathbb{P}(A|Z_0=j)$ is a constant for all $j$? How does recurrence of this markov chain help us to prove it?

2. If $A$ is any event, does (1) still hold true?

Thanks in advance!