# Kolmogorov’s extension theorem

For all $t_{1},\cdots,t_{k}$, $k\in\mathbb{N}$, let $v_{t_{1},\cdots,t_{k}}$ be probability measures  on $\mathbb{R}^{nk}$ satisfying the following properties (consistency conditions):

1. 1.

$v_{t_{\sigma(1)},\cdots,t_{\sigma(k)}}(F_{1}\times\cdots\times F_{k})=v_{t_{1}% ,\cdots t_{k}}(F_{\sigma^{-1}(1)}\times\cdots F_{\sigma^{-1}(k)})$ for all permutations  $\sigma$ of $\{1,2,\cdots,k\}$ and for all Borel sets $F_{i}$ of $\mathbb{R}^{n}$

2. 2.

$v_{t_{1},\cdots,t_{k}}(F_{1}\times\cdots\times F_{k})=v_{t_{1},\cdots,t_{k},t_% {k+1},\cdots t_{k+m}}(F_{1}\times\cdots\times F_{k}\times\mathbb{R}^{n}\times% \cdots\times\mathbb{R}^{n})$ for all $m\in\mathbb{N}$ and for all Borel sets $F_{i}$ of $\mathbb{R}^{n}$

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

 $v_{t_{1},\cdots,t_{k}}(F_{1}\times\cdots\times F_{k})=P(X_{t_{1}}\in F_{1},% \cdots,X_{t_{k}}\in F_{k})$

for all $t_{i}\in T,k\in\mathbb{R}^{n}$ and all Borel sets $F_{i}$ of $\mathbb{R}^{n}$

Title Kolmogorov’s extension theorem KolmogorovsExtensionTheorem 2013-04-12 21:33:32 2013-04-12 21:33:32 Filipe (28191) Filipe (28191) 3 Filipe (28191) Theorem msc 60G07