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 $p_{ij}$?