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!
The Kolmogorov Extension Theorem may be able to help. The "Implications of the Theorem" section seems relevant to your question here. It's not constructive at the infinite level, but provides a formalism to link the finite to the infinite.