It is credited to the English mathematician Percy John Daniell and the Russian mathematician Andrey Nikolaevich Kolmogorov.
denote some interval (thought of as "time"), and let
Suppose that these measures satisfy two consistency conditions: 1. for all permutations
as its finite-dimensional distributions relative to times
In fact, it is always possible to take as the underlying probability space
Therefore, an alternative way of stating Kolmogorov's extension theorem is that, provided that the above consistency conditions hold, there exists a (unique) measure
Kolmogorov's extension theorem applies when
is uncountable, but the price to pay for this level of generality is that the measure
The two conditions required by the theorem are trivially satisfied by any stochastic process.
The first condition generalizes this statement to hold for any number of time points
Also this is a trivial condition that will be satisfied by any consistent family of finite-dimensional distributions.
Since the two conditions are trivially satisfied for any stochastic process, the power of the theorem is that no other conditions are required: For any reasonable (i.e., consistent) family of finite-dimensional distributions, there exists a stochastic process with these distributions.
The measure-theoretic approach to stochastic processes starts with a probability space and defines a stochastic process as a family of functions on this probability space.
However, in many applications the starting point is really the finite-dimensional distributions of the stochastic process.
The theorem says that provided the finite-dimensional distributions satisfy the obvious consistency requirements, one can always identify a probability space to match the purpose.
In many situations, this means that one does not have to be explicit about what the probability space is.
Many texts on stochastic processes do, indeed, assume a probability space but never state explicitly what it is.
The theorem is used in one of the standard proofs of existence of a Brownian motion, by specifying the finite dimensional distributions to be Gaussian random variables, satisfying the consistency conditions above.
As in most of the definitions of Brownian motion it is required that the sample paths are continuous almost surely, and one then uses the Kolmogorov continuity theorem to construct a continuous modification of the process constructed by the Kolmogorov extension theorem.
The Kolmogorov extension theorem gives us conditions for a collection of measures on Euclidean spaces to be the finite-dimensional distributions of some
-valued stochastic process, but the assumption that the state space be
In fact, any collection of measurable spaces together with a collection of inner regular measures defined on the finite products of these spaces would suffice, provided that these measures satisfy a certain compatibility relation.
The formal statement of the general theorem is as follows.
which is inner regular with respect to the product topology (induced by the
of measures satisfies the following compatibility relation: for finite subsets
Then there exists a unique probability measure
are defined on the product sigma algebra on their respective spaces, which (as mentioned before) is rather coarse.
may sometimes be extended appropriately to a larger sigma algebra, if there is additional structure involved.
The reason that the original statement of the theorem does not mention inner regularity of the measures
This theorem has many far-reaching consequences; for example it can be used to prove the existence of the following, among others: According to John Aldrich, the theorem was independently discovered by British mathematician Percy John Daniell in the slightly different setting of integration theory.