Probability theory and stochastic processes Independence Theorem

Ionescu-Tulcea theorem

Builds on Markov kernel Probability measure Product sigma-algebra Measurable space

Statement

For each $i \in \set{0, 1, 2, \ldots}$ let $(\Omega_i, \mathcal{F}_i)$ be a measurable space, and for $n \ge 0$ put $\Omega^n = \Omega_0 \times \cdots \times \Omega_n$ with the product σ-algebra. Let $P_0$ be a probability measure on $\Omega_0$ . For each $n \ge 1$ let $\kappa_n$ be a Markov kernel from $\Omega^{n-1}$ to $\Omega_n$ . Then 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 $A_i \in \mathcal{F}_i$ , $P\Big( A_0 \times \cdots \times A_n \times \prod_{i > n} \Omega_i \Big) = \int_{A_0} \int_{A_1} \cdots \int_{A_n} \kappa_n(\omega_0, \ldots, \omega_{n-1}, d\omega_n) \cdots \kappa_1(\omega_0, d\omega_1) \, P_0(d\omega_0).$

Remarks

The 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. Countable additivity is proved by contradiction. Suppose a decreasing sequence of cylinder sets has measures bounded below by $\varepsilon > 0$ and empty intersection. Using the kernels, one chooses coordinates $\omega_0, \omega_1, \ldots$ one at a time and gets a point in all of the sets. Carathéodory's extension theorem then gives $P$ .

Example: 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 $S$ . Then $P$ is the law of a Markov chain on $S$ with initial law $P_0$ and transition kernel $K$ .

Unlike the Kolmogorov extension theorem, this theorem needs no topological assumption on the spaces $\Omega_i$ . The price is that the data must come as kernels, that is, as conditional laws given the past, and the index set is countable.

Going deeper

For a student

The theorem is the general existence result for discrete-time processes, in particular Markov chains on general state spaces and controlled processes in dynamic programming. With kernels that ignore the past it gives the countable infinite product measure. It 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.

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Tag	`94ASD187AQ`
URI	`m2://94ASD187AQ`
Kind	statement, theorem
Writing	W0 stub
MSC	60J05, 28A35, 60A10
Status	This copy of the site has no build data. Run `m2 status 94ASD187AQ`.
About	Markov kernel, Probability measure
Uses	Markov kernel, Probability measure, Product sigma-algebra, Measurable space
Same as	Wikidata Q25378690 Wikipedia Ionescu-Tulcea theorem
Changed	2 November 2026

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