Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space. I want to prove the uniqueness of a probability distribution under which a process $(X_n)_{n\geq0}$ is a Markov chain with transition probabilities $P$ and initial distribution $\mu_0$.
As an hint I have that I have to use Dynkin's lemma. In order to do this I write the statement below:
Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space. Let $\alpha$ be the statement which I want to prove, and which should hold for some set in $\Omega$. Let $\Sigma$ a $\sigma$-algebra with a generator stable under intersections for which $\alpha$ holds. Then check that $\mathcal{D}:=\{A\in \Sigma\mid A \text{ satisfies } \alpha\}$ is a Dynkin's system.
Let now in our situation $P_{\mu_0}^1,P_{\mu_0}^2$ be two probability distributions under which the process $(X_n)_{n\geq 0}$ is a Markov chain with transition probabilities $P$ and initial distribution $\mu_0$. We already know that $\forall x_0,...,x_n\in E$, $E$ is the state space of the chain, $P_{\mu_0}^i=\mu_0(x_0)p_{x_0x_1}\dots p_{x_{n-1}x_n}$ for $i=1,2$. I took $\Sigma$ to be the sigma algebra of the probability space where my Markov chain is defined. I have troubles to find my generating set of the sigma-algebra $\Sigma=\mathcal{F}$.
thanks for any help.
The phrase "the sigma algebra of the probability space where my Markov chain is defined" is pretty vague. Specifically, you need to use the $\sigma$-algebra generated by the process, i.e., ${\cal F}=\sigma(X_0,X_1,\dots).$ A generating set $\cal D$ for $\cal F$ is the collection of all sets of the form $$\bigcap_{j=0}^n\,\{\omega\in \Omega: X_j(\omega)=x_j\}.$$ As you know, both measures $P^1_{\mu_0}$ and $P^2_{\mu_0}$ assign the same probability to such sets, so you are done!