A Markov chain $(X_i)_{i\in \mathbb{N}}$ on a measurable space $(E,\Sigma)$ is (see e.g. Revuz or Meyn/Tweedie) constructed on the following probabilty space. $$ \Omega = \{ (x_l)_{l \in \mathbb{N}} \mid x_l \in E \text{ for all } l \in \mathbb{N} \}$$ and $X_i$ is definied as $$X_i ((x_l)_{l \in \mathbb{N}}) = x_i$$ and $\Sigma$ is constructed suitable. Then one can show, for any Markov kernel p and probability measure $\mu$ exists a a probability measure $\mathbb{P}_\mu$ on $\Omega$ with $$ \mathbb{P}_\mu [X_0 \in A_0, X_1 \in A_1 ,\dots X_n \in A_n] =$$ $$ \int_{A_0} \dots \int_{A_n} p(y_{n-1},A_n) \, p(y_{n-2}, dy_{n-1}) \dots p(y_0, dy_1) \, \mu(dy_0).$$ Thus by this particular construction for all Markov chains the space $(\Omega,\Sigma)$ and the random variable $(X_i)_{i \in \mathbb{N}}$ are identical and two Markov chains only differ in the associated probability measure $\mathbb{P}_\mu$.

**My question:**
Assume two measures $\mu_1$ and $\mu_2$ and two Markov kernels $p_1$ and $p_2$ are given. Then I have two seperate probability measures $\mathbb{P}_{\mu_1}$ and $\mathbb{P}_{\mu_2}$ where the projection process lives on.
Can I conclude from this that there exits a single probability space $(\hat{\Omega},\hat{\Sigma},\mathbb{P})$ together with two stochastic processes $(Z_i)$ and $(Y_i)$ with $$ \mathbb{P} [Z_0 \in A_0, Z_1 \in A_1 ,\dots Z_n \in A_n] = \mathbb{P}_{\mu_1} [X_0 \in A_0, X_1 \in A_1 ,\dots X_n \in A_n]$$

and $$ \mathbb{P} [Y_0 \in A_0, Y_1 \in A_1 ,\dots Y_n \in A_n] = \mathbb{P}_{\mu_2} [X_0 \in A_0, X_1 \in A_1 ,\dots X_n \in A_n]?$$