---
tag: 94ASD187AQ
kind: statement
role: theorem
name: Ionescu-Tulcea theorem
msc: [60J05, 28A35, 60A10]
uses: [2AVNDP8BJP, 059TJN0NZ1, 6HVCTJGB61, N4KDT83V3S]
about: [2AVNDP8BJP, 059TJN0NZ1]
sources:
  - url: https://en.wikipedia.org/wiki/Ionescu-Tulcea_theorem
same_as:
  wikidata: Q25378690
  wikipedia: Ionescu-Tulcea theorem
---

## Statement

For 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]].
Let $P_0$ be a [[probability measure|059TJN0NZ1]] on $\Omega_0$.
For each $n \ge 1$ let $\kappa_n$ be a [[Markov kernel|2AVNDP8BJP]] 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|Q8EXG7EKYW]] 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|WR4X7MMTH2]], 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.

## Explanations

### L1 human

Think of a random process that runs step by step.
At each step, the law of the next value may depend on everything that happened before.
The theorem says that such step-by-step rules always define a single probability measure on the space of whole infinite histories.

### L2 human

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|2HWVFHM13R]].
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.