I recently started reading Wolfgang Woess' book titled "Random Walks on Infinite Groups". In the section where he introduces Markov chains and random walks on a set $X$, he has defined a probability measure $\mathbb{P}_x$ on $X^{\mathbb{N}_0}$ corresponding to a Markov chain $(Z_n)_{n\geq 0}$ starting at $x\in X$. He has also written how $\mathbb{P}_x$ operates:
$$\mathbb{P}_x[Z_n=x_0, Z_1=x_1,....,Z_n=x_n]=\delta_x(x_0)p(x_0,x_1)\dots p(x_{n-1}, x_n)$$
where $(p(x,y))_{x,y\in X}$ is the transition matrix of the random walk $(Z_n)_{n\geq 0}$.
Question 1: I don't really understand how to apply $\mathbb{P}_x$ on a general measurable set from this given information. For instance, how to calculate $\mathbb{P}_x(Z_m=z)$? More generally how to calculate $\mathbb{P}_x(Y=y)$ where $Y:X^{\mathbb{N}_0}\rightarrow X$ is a random variable ?
Question 2: How does $\mathbb{P}_x$ relate to $(Z_n)_{n\geq 0}$ as $x$ varies?
I am a novice in probability theory so I am extremely sorry in advance if some of my doubts seem absurd. Thanks in advance!
