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  "body": "\n## Statement\n\nFor each $i \\in \\set{0, 1, 2, \\ldots}$ let $(\\Omega_i, \\mathcal{F}_i)$ be a [[measurable space|N4KDT83V3S]], and for $n \\ge 0$ put $\\Omega^n = \\Omega_0 \\times \\cdots \\times \\Omega_n$ with the [[product σ-algebra|6HVCTJGB61]].\nLet $P_0$ be a [[probability measure|059TJN0NZ1]] on $\\Omega_0$.\nFor each $n \\ge 1$ let $\\kappa_n$ be a [[Markov kernel|2AVNDP8BJP]] from $\\Omega^{n-1}$ to $\\Omega_n$.\nThen there is a unique probability measure $P$ on $\\prod_{i=0}^{\\infty} \\Omega_i$ with its product σ-algebra such that, for all $n \\ge 0$ and all $A_i \\in \\mathcal{F}_i$,\n$$P\\Big( A_0 \\times \\cdots \\times A_n \\times \\prod_{i \u003e n} \\Omega_i \\Big) = \\int_{A_0} \\int_{A_1} \\cdots \\int_{A_n} \\kappa_n(\\omega_0, \\ldots, \\omega_{n-1}, d\\omega_n) \\cdots \\kappa_1(\\omega_0, d\\omega_1) \\, P_0(d\\omega_0).$$\n\n## Remarks\n\nThe right side defines a consistent family of probability measures $P^n$ on the spaces $\\Omega^n$, so it gives a finitely additive set function on cylinder sets.\nCountable additivity is proved by contradiction.\nSuppose a decreasing sequence of cylinder sets has measures bounded below by $\\varepsilon \u003e 0$ and empty intersection.\nUsing the kernels, one chooses coordinates $\\omega_0, \\omega_1, \\ldots$ one at a time and gets a point in all of the sets.\n[[Carathéodory's extension theorem|Q8EXG7EKYW]] then gives $P$.\n\nExample: let $\\Omega_i = S$ for a measurable space $S$, let $P_0$ be an initial law, and let $\\kappa_n(\\omega_0, \\ldots, \\omega_{n-1}, \\cdot) = K(\\omega_{n-1}, \\cdot)$ for one Markov kernel $K$ on $S$.\nThen $P$ is the law of a Markov chain on $S$ with initial law $P_0$ and transition kernel $K$.\n\nUnlike the [[Kolmogorov extension theorem|WR4X7MMTH2]], this theorem needs no topological assumption on the spaces $\\Omega_i$.\nThe price is that the data must come as kernels, that is, as conditional laws given the past, and the index set is countable.\n\n## Explanations\n\n### L1 human\n\nThink of a random process that runs step by step.\nAt each step, the law of the next value may depend on everything that happened before.\nThe theorem says that such step-by-step rules always define a single probability measure on the space of whole infinite histories.\n\n### L2 human\n\nThe theorem is the general existence result for discrete-time processes, in particular Markov chains on general state spaces and controlled processes in dynamic programming.\nWith kernels that ignore the past it gives the countable [[infinite product measure|2HWVFHM13R]].\nIt complements the Kolmogorov extension theorem: Kolmogorov needs Polish spaces and handles any index set, while Ionescu-Tulcea needs kernels and handles arbitrary measurable spaces with a countable sequence of steps.\n",
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