Consider a Markov Chain $(X,\mathcal{F},\mu,T)$, where $X = (1,2,\dots,n)^\mathbb{Z}$ and $T$ is the left shift and a transition matrix $P=(p_{i,j})$ and stationary distribution $\pi$ such that $\pi P = \pi$, with the Markov measure defined as:
$$ \mu(\{x \in X: x_{-k} = a_{-k},\dots,x_k = a_k\}) = \pi_{a_{-k}}p_{a_{-k}a_{-k+1}}\dots p_{a_{k-1}a_{k}}$$
In all the literature I've found on the matter, it is stated that extends to the full $\sigma$-algebra by Kolmogorov's Extension theorem. My problem is, I don't understand how this extends to cylinders where only a single coordinate is specified, for eg:
$$ \mu(\{x \in X: x_0=i\}) = \,?$$
Is it just $\pi_i$, $\pi_i p_{ii}$, or some summation of $p_{ij}$?
The definition is $$\mu\{x\in X,x_j=a_j,-k\leqslant j\leqslant k\}:=\pi_{a_{-k}}\prod_{j=-k}^{k-1}p_{a_ja_{j+1}}.$$ When $k=0$, the set of indexes of the product is empty and by convention the product is $1$, so $\mu\{x\in X,x_0=i\}=\pi_{i}$.