Kolmogorov extension theorem
In mathematics, the Kolmogorov extension theorem (also known as Kolmogorov existence theorem, the Kolmogorov consistency theorem or the Daniell-Kolmogorov theorem) is a theorem that guarantees that a suitably "consistent" collection of finite-dimensional distributions will define a stochastic process. It is credited to the English mathematician Percy John Daniell and the Russian mathematician Andrey Nikolaevich Kolmogorov. Statement of the theorem. Let formula_1 denote some interval (thought of as "time"), and let formula_2. For each formula_3 and finite sequence of distinct times formula_4, let formula_5 be a probability measure on formula_6. Suppose that these measures satisfy two  consistency conditions: 1. for all permutations formula_7 of formula_8 and measurable sets formula_9, 2. for all measurable sets formula_9,formula_12 Then there exists a probability space formula_14 and a stochastic process formula_15 such that for all formula_17, formula_3 and measurable sets formula_9, i.e. formula_20 has formula_5 as its finite-dimensional distributions relative to times formula_22. In fact, it is always possible to take as the underlying probability space formula_23 and to take for formula_20 the canonical process formula_25. Therefore, an alternative way of stating Kolmogorov's extension theorem is that, provided that the above consistency conditions hold, there exists a (unique ) measure formula_26 on formula_27 with marginals formula_5 for any finite collection of times formula_22. Kolmogorov's extension theorem applies when formula_1 is uncountable, but the price to pay for this level of generality is that the measure formula_26 is only defined on the product σ-algebra of formula_27, which is not very rich. Explanation of the conditions. The two conditions required by the theorem are trivially satisfied by any stochastic process. For example, consider a real-valued discrete-time stochastic process formula_20. Then the probability formula_34 can be computed either as formula_35 or as formula_36. Hence, for the finite-dimensional  distributions to be consistent, it must hold that formula_37. The first condition generalizes this statement to hold for any number of time points formula_38, and any control sets formula_39. Continuing the example, the second condition implies that formula_40. Also this is a trivial condition that will be satisfied by any consistent family of finite-dimensional distributions. Implications of the theorem. 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. General form of the theorem. The Kolmogorov extension theorem gives us conditions for a collection of measures on Euclidean spaces  to be the finite-dimensional distributions of some formula_41-valued stochastic process, but the assumption that the state space be formula_41 is unnecessary. 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. Let formula_1 be any set. Let formula_44 be some collection of measurable spaces, and for each formula_45, let formula_46 be a Hausdorff topology on formula_47. For each finite subset formula_48, define For subsets  formula_50, let formula_51 denote the canonical projection map formula_52. For each finite subset formula_53, suppose we have a probability measure formula_54 on formula_55 which is inner regular with respect to the product topology (induced by the formula_56) on formula_57. Suppose also that this collection formula_58 of measures satisfies the following compatibility relation: for finite subsets formula_59, we have that where formula_61 denotes the pushforward measure of formula_62 induced by the canonical projection map formula_63. Then there exists a unique probability measure formula_64 on formula_65 such that formula_66 for every finite subset formula_67. As a remark, all of the measures formula_68  are defined on the product sigma algebra on their respective spaces, which (as mentioned before) is rather coarse. The measure formula_64 may sometimes be extended appropriately to a larger sigma algebra, if there is additional structure involved. Note that the original statement of the theorem is just a special case of this theorem with formula_70 for all formula_71, and formula_72 for formula_73. The stochastic process would simply be the canonical process formula_74, defined on formula_75 with probability measure formula_76. The reason that the original statement of the theorem does not mention inner regularity of the measures formula_77 is that this  would automatically follow, since Borel probability measures on Polish spaces are automatically Radon. This theorem has many far-reaching consequences; for example it can be used to prove the existence of the following, among others: History. According to John Aldrich, the theorem was independently discovered by British mathematician Percy John Daniell in the slightly different setting of integration theory.