Kolmogorov extension theorem

In mathematics, the Kolmogorov extension theorem (also known as Kolmogorov existence theorem or Kolmogorov consistency theorem) is a theorem that guarantees that a suitably "consistent" collection of finite-dimensional distributions will define a . It is credited to the Soviet mathematician Andrey Nikolaevich Kolmogorov.

Let ${\displaystyle T}$ denote some interval (thought of as "time"), and let ${\displaystyle n\in \mathbb {N} }$. For each ${\displaystyle k\in \mathbb {N} }$ and finite sequence of distinct times ${\displaystyle t_{1},\dots ,t_{k}\in T}$, let ${\displaystyle \nu _{t_{1}\dots t_{k}}}$ be a probability measure on ${\displaystyle (\mathbb {R} ^{n})^{k}}$. Suppose that these measures satisfy two consistency conditions:

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