Kolmogorov's existence theorem
/ kol-mo-GO-rov /
How do you know a Gaussian process — or any stochastic process indexed by a continuum of times — actually exists as a single random object living on one probability space, rather than just a wish list of finite-dimensional distributions? Kolmogorov's existence (or extension) theorem is the answer: it manufactures the process from a consistent family of finite-dimensional laws. It is the licence to write down a covariance function and declare 'let X be the Gaussian process with this covariance.'
Set up a state space (say R or R^d) and, for every finite ordered list of indices t_1, ..., t_n, a probability measure mu_(t_1, ..., t_n) on R^n meant to be the joint law of (X_(t_1), ..., X_(t_n)). The theorem says: if this family satisfies two natural consistency (Kolmogorov consistency) conditions — symmetry under reordering the indices, and marginalization (dropping an index and integrating it out gives the lower-dimensional measure) — then there exists a probability measure P on the product space R^T (sequences/functions indexed by T) whose finite-dimensional marginals are exactly the prescribed mu's. For a Gaussian process the finite-dimensional laws are the multivariate normals N(m restricted, K restricted) read off from a positive-semidefinite K, and consistency is automatic, so existence is guaranteed.
Why it matters and what it does NOT give: the theorem builds the law on the product sigma-algebra, where events depend on only countably many coordinates. This is enough to define the process but NOT enough to control path properties like continuity — the set of continuous functions is not even measurable in the product sigma-algebra. So existence comes first (Kolmogorov), then a SEPARATE argument (Kolmogorov's continuity theorem, or Gaussian sample-path criteria) is needed to find a version with nice (e.g. continuous) paths. A second caveat: the underlying state space must be reasonable (e.g. Polish / standard Borel) for the construction to go through unconditionally.
To build Brownian motion, prescribe for t_1 < ... < t_n the joint law in which the increments B_(t_1), B_(t_2) - B_(t_1), ... are independent Gaussians with variances t_1, t_2 - t_1, .... These laws are consistent, so Kolmogorov hands you a process on R^[0, infinity); a second theorem (continuity) then upgrades it to a version with continuous paths.
Kolmogorov's theorem turns a consistent family of finite-dimensional laws into an actual process; path regularity is a separate, later step.
Existence does not bring continuity: continuous (or càdlàg) paths require a separate version theorem, since such path sets are non-measurable in the product sigma-algebra.