I'm reading Continuous Time Markov Processes: An Introduction by Thomas M. Liggett. Chapter 2.4 is devoted to Blackwell's example.
Let $E=\left\{0,1\right\}$, $\mathcal E:=2^E$ and $X$ be the (Question 1: "Is $X$ really uniquely determined?") Markov process with values in $(E,\mathcal E)$ and $Q$-matrix (also called transition rate matrix) $$\left(\begin{matrix}-\beta&\beta\\\delta&-\delta\end{matrix}\right)$$ for some $\beta,\delta>0$.
Now, let $\left\{\beta_n\right\},\left\{\delta_n\right\}\subseteq(0,\infty)$ and $\left\{X_n\right\}$ be an independent (Question 2: "Why is it possible to guarantee independence?") family of two-state Markov proceesses according to $X$ above.
Now, let $\pi_n:E^{\mathbb N}\to E$ be the $n$-th coordinate map and define $Y$ by $$\pi_n\circ Y=X_n\;,$$ i.e. $Y$ is the process with values in $\left(E^{\mathbb N_0},\mathcal E^{\otimes\mathbb N}\right)$ defined by $$Y(t):=\left(X_1(t),X_2(t),\ldots\right)\;\;\;\text{for }t\ge 0\;.$$
Liggett states, that $Y$ "is well-defined by the Kolmogorv extension theorem", which can be stated as follows:
Let $(\Omega_i,\mathcal A_i)$ be a Borelian measurable space and $\left(\operatorname P_J:J\subseteq I\text{ finite}\right)$ be a projective family of probability measures on $$\left(\Omega_J,\mathcal A_J\right):=\left(\prod_{j\in J}\Omega_j,\bigotimes_{j\in J}\mathcal A_j\right)\;.$$ Then, there is a unique probability measure $\operatorname P$ on $(\Omega_I,\mathcal A_I)$ with $$\operatorname P_J=\operatorname P\circ\pi_J^{-1}\;\;\;\text{for all finite }J\subseteq I\;,$$ where $\pi_J:\Omega_I\to\Omega_J* are the canonical projections.
Question 3: I don't understand why we need the extension theorem and why it guarantees well-definedness. Why should $Y$ not be well-defined, at all?