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 . Then the probability can be computed either as or as . Hence, for the finite-dimensional distributions to be consistent, it must hold that . The first condition generalises this obvious statement to hold for any number of time points, and any control sets .
Continuing the example, the second condition implies that . Also this is a trivial statement that must be satisfied for any consistent family of finite-dimensional distributions.
Read more about this topic: Kolmogorov Extension Theorem
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