nLab Kolmogorov extension theorem

Idea

One of the main tenets of probability theory is that in general the probability of the product is not the product of the probabilities, because of correlation and other statistical interactions. In particular, the probability of an infinite product is not the infinite product of the probabilities.

However, an infinite product is a cofiltered limit of finite products, and under some conditions, the probability of an infinite product is the limit of the probability of the finite products. One can interpret this fact as the absence of “infinitary correlations”.

Statements of this kind are known collectively as the Kolmogorov extension theorem. They are particularly important in the theory of stochastic processes, since a stochastic process can be seen as a joint distribution over an infinite product.

In terms of category theory, this can be phrased as the fact that this cofiltered limit is preserved by the left adjoint from measurable spaces to Markov kernels (or equivalently, by the Giry monad).

In measure-theoretic probability

We look at the classical statement, and then give two (related) ways to express it category-theoretically.

Classical statement

The classical statement is about probability measures, and roughly says that under some conditions, a probability measure on an infinite product is uniquely determined by its finite marginals.

Theorem

Let $\{(X_j,\mathcal{A}_j)\}_{j\in J}$ be a collection of standard Borel spaces, indexed by a set $J$ of arbitrary cardinality.

• Denote by $(X_J,\mathcal{A}_J)$ the product of the $X_j$, equipped with the product sigma-algebra.
• Similarly, for every finite set $F\subseteq J$, denote $(X_F,\mathcal{A}_F)$ the product of the $(X_j,\mathcal{A}_j)$ for $j\in F$.

There is a bijection between

• probability measures on $(X_J,\mathcal{A}_J)$

and

• families of probability measures $p_F$ on $(X_F,\mathcal{A}_F)$ for each finite subset $F\subseteq J$ such that for every inclusion $F\subseteq G$, the measure $p_F$ is the marginal of $p_G$.

In particular, the $p_F$ can be obtained as marginal distributions of the measure $p_J$.

Note that

• The same result holds more in general if each $(X_j,\mathcal{A}_j)$ is a Hausdorff space equipped with its Borel sigma-algebra, and the measures $p_F$ are inner regular?.

• The uncountable product of standard Borel spaces may fail to be standard Borel.

• The Borel sigma-algebra of an uncountable product of topological spaces is not the product of the Borel sigma-algebras of the factors.

In the category of Markov kernels

We can restate Kolmogorov’s extension theorem as a limit, as follows.

Let $\{(X_j,\mathcal{A}_j)\}_{j\in J}$ be a collection of standard Borel spaces, indexed by a set $J$ of arbitrary cardinality.

• Denote by $(X_J,\mathcal{A}_J)$ the categorical product of the $X_j$ in the category Meas of measurable spaces and measurable maps.
• Similarly, for every finite set $F\subseteq J$, denote $(X_F,\mathcal{A}_F)$ the product of the $(X_j,\mathcal{A}_j)$ for $j\in F$.

Note that, since infinite products are cofiltered limits of finite products, $(X_J,\mathcal{A}_J)$ is the limit in Meas of the diagram formed by the $(X_F,\mathcal{A}_F)$ and their projections.

Now:

Corollary

The object $(X_J,\mathcal{A}_J)$ is the limit of the $(X_F,\mathcal{A}_F)$ also in the category Stoch of measurable spaces and Markov kernels.

Note that this does not say that $(X_J,\mathcal{A}_J)$ is the product of the $(X_j,\mathcal{A}_j)$ (that would be false even in the finite case). It is just the “finite to infinite” part which is retained in the category of kernels.

Proof

For finite sets $F\subseteq G\subseteq J$, denote by $\pi_{J,F}:(X_J,\mathcal{A}_J)\to(X_F,\mathcal{A}_F)$ and $\pi_{G,F}:(X_G,\mathcal{A}_H)\to(X_F,\mathcal{A}_F)$ the product projections. With a slight abuse, denote the Markov kernel induced by the functions $\pi_{J,F}$ and $\pi_{G,F}$ again by $\pi_{J,F}$ and $\pi_{G,F}$.

Given a measurable space $(Y,\mathcal{B})$, we have to show that there is a bijection between Markov kernels $k_J:Y\to X_J$ and families of Markov kernels $k_F:Y\to X_F$ such that for each inclusion $F\subseteq G$, the outer triangle in the following diagram of Stoch commutes.

Now for each $y\in Y$, the Markov kernels $k_J(-|y)$ and $k_F(-|y)$ are just probability measures, so that Theorem applies. We only need to show that $k_J$ is measurable in $y$ if and only if the $k_F$ are. Now if $k_J$ is measurable, so is $k_F$, since the composition of measurable functions is measurable. The converse is true as well, since measurability of Markov kernels can be equivalently tested on a pi-system?.

In terms of the Giry monad

The result in terms of Markov kernels can be restated in terms of the Giry monad as follows.

Once again, recall that $(X_J,\mathcal{A}_J)$ is the cofiltered limit in Meas of the diagram formed by the $(X_F,\mathcal{A}_F)$ and their projections.

Now, if the $X_j$ are standard Borel:

Corollary

The Giry monad preserves this limit.

Proof

We have to show that given a measurable space $Y$, there is a bijection between measurable functions $f_J:Y\to X_J$ and families of measurable functions $f_F:Y\to X_F$ such that for each inclusion $F\subseteq G$, the outer triangle in the following diagram of Meas commutes. Now since Stoch is the Kleisli category of the Giry monad, notice that there is a bijection

$$Stoch(Y,X) \cong Meas(Y,PX)$$

natural in $X$, so that the universal property that we want to prove is just a restatement of the proof of Corollary .

In Markov categories

In Markov categories the Kolmogorov extension theorem appears in the form of an axiom that the category can satisfy, an abstraction of the universal property in the category of Markov kernels given above.

Definition

Let $C$ be a Markov category, and let $(X_j)_{j\in J}$ be a family of objects. A Kolmogorov product of them is an infinite tensor product

$$X_J=\bigotimes_{j\in J} X_j$$

such that all finite projections (marginalizations) $X_J\to X_F$, with $F\subseteq J$ finite, are deterministic morphisms.

The category BorelStoch, for example, has all countable Kolmogorov products. (The uncountable product of standard Borel spaces, in general, is not standard Borel.)

See Markov category - Kolmogorov products for more on this, as well as Fritz-Rischel’20.

See also

• probability theory, categorical probability
• filtered category, filtered limit
• infinitary tensor product
• probability monad, Giry monad
• Markov category, infinitary tensor product
• joint and marginal distributions
• stochastic dependence and independence

References

• Tobias Fritz and Eigil Fjeldgren Rischel, Infinite products and zero-one laws in categorical probability, Compositionality 2(3) 2020. (arXiv:1912.02769)

• Ruben Van Belle, A categorical treatment of the Radon-Nikodym theorem and martingales, 2023. (arXiv)

category: probability