# How to define measure over infinite sequence of random variables

Suppose we are drawing elements from a set $$\Omega$$ at each time $$t\in \mathbb N$$. The probability of drawing $$\omega\in \Omega$$ at time $$t$$ is given by a conditional distribution $$\mu(\omega|\omega_1,\omega_2,\ldots,\omega_{t-1})$$ (the function $$(\omega_1,\omega_2,\ldots,\omega_{t-1})\to \mu(B|\omega_1,\omega_2,\ldots,\omega_{t-1})$$ is measurable in $$\Omega^{t-1}$$ for each $$B\subseteq \Omega$$). How does one formally define the joint probability measure over a realization of an infinite sequence of draws such that the probability of any finite sequence of draws corresponds to the one given by the joint measure induced by the $$\mu$$'s?

Does anyone have a reference for this type of construction?

Thanks!