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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!

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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.

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