Kolmogorov extension theorem

The Kolmogorov extension theorem guarantees a probability measure on an infinite product space when a family of finite-dimensional distributions satisfies consistency conditions.

Projective consistency: Compatibility of distributions under maps that discard coordinates from a larger finite-dimensional space. Marginalizing any specified coordinates must recover the distribution assigned to those remaining.
Probability measure: A measure assigning total mass one to a measurable space and assigning probabilities to its measurable sets. The theorem's conclusion is precisely a probability measure on the product space.
Wiener measure: The probability measure on continuous paths that describes standard Brownian motion. Brownian motion can be characterized through compatible finite-dimensional Gaussian distributions.
Daniell–Kolmogorov theorem: A formulation of the extension theorem using positive linear functionals on spaces of continuous functions. It offers an alternative framework for extending finite-dimensional information.
Marginal distribution: The probability distribution of a subset of variables obtained by integrating out the others. Consistency is expressed by requiring larger joint laws to have the prescribed smaller marginals.
Measurable space: A set equipped with a sigma-algebra that specifies which subsets can be measured. Each coordinate space must carry measurable structure before finite-dimensional laws can be stated.
Brownian motion: A continuous-time stochastic process with independent Gaussian increments and continuous sample paths. Its finite-dimensional laws determine a process law through extension, with path properties requiring further arguments.
Carathéodory extension theorem: A theorem that extends a suitably defined premeasure to a measure on a generated sigma-algebra. This general measure-extension result supplies a method, while Kolmogorov specifies compatible probability marginals.
Cylinder set: A subset of a product space whose membership depends on only finitely many coordinates. Finite-dimensional probabilities first specify probabilities for these basic events.
Sigma-algebra: A collection of subsets closed under complements and countable unions, including the whole space. The theorem's probability measure is defined on a sigma-algebra generated by coordinate events.

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