The Kolmogorov extension theorem states that a consistent family of finite-dimensional probability distributions determines
a probability measure on the corresponding
infinite product space with its product sigma-algebra. More precisely, let 
be an index set and let 
be a standard Borel space. Suppose that for every finite 
a probability measure 
on 
is given, and that whenever 
, the coordinate projection from 
to 
sends 
to 
. Then there is a unique probability measure 
on 
whose marginal on every 
is 
.
The theorem provides a basic construction of stochastic processes: specifying compatible joint distributions at finitely many times determines a process on the full index set.
