Kolmogorov’s extension theorem

For all $t_1,\mathrm{\cdots },t_k$, $k\in \mathbb{N}$, let $v_{t_1,\mathrm{\cdots },t_k}$ be probability measures on ${ℝ}^{nk}$ satisfying the following properties (consistency conditions):

1. $v_{t_{\sigma (1)},\mathrm{\cdots },t_{\sigma (k)}}(F_1\times \mathrm{\cdots }\times F_k)=v_{t_1,\mathrm{\cdots }{t}_k}(F_{{\sigma }^{-1}(1)}\times \mathrm{\cdots }{F}_{{\sigma }^{-1}(k)})$ for all permutations $\sigma$ of $\{1,2,\mathrm{\cdots },k\}$ and for all Borel sets $F_i$ of ${ℝ}^n$

2. $v_{t_1,\mathrm{\cdots },t_k}(F_1\times \mathrm{\cdots }\times F_k)=v_{t_1,\mathrm{\cdots },t_k,t_{k+1},\mathrm{\cdots }{t}_{k+m}}(F_1\times \mathrm{\cdots }\times F_k\times {ℝ}^n\times \mathrm{\cdots }\times {ℝ}^n)$ for all $m\in \mathbb{N}$ and for all Borel sets $F_i$ of ${ℝ}^n$

Then there exists a probability space $(\mathrm{\Omega },ℱ,P)$ and a stochastic process $X_t$ on $\mathrm{\Omega }$, indexed by $T$, taking values in ${ℝ}^n$ such that

$$v_{t_1,\mathrm{\cdots },t_k}(F_1\times \mathrm{\cdots }\times F_k)=P(X_{t_1}\in F_1,\mathrm{\cdots },X_{t_k}\in F_k)$$

for all $t_i\in T,k\in {ℝ}^n$ and all Borel sets $F_i$ of ${ℝ}^n$