In Økensdal's Chapter 2 (6th ed. of Introduction to Stochastic Differential Equations), the author states the Kolmogorov's Extension Theorem as follows:
For $k$ Borel sets $F_1, \cdots, F_k \in \mathbb{R}^n$, for all $t_1, \cdots, t_k \in T$, $k \in \mathbb{N}$, let $\nu_{t_1},\cdots,t_k$ be probability measures on $\mathbb{R}^{nk}$ such that $$ \nu_{t_{\sigma(1)},\cdots,t_{\sigma(k)}}\left(F_1 \times \cdots \times F_k\right) = \nu_{t_1,\cdots,t_k}\left(F_{\sigma^{-1}(1)} \times \cdots \times F_{\sigma^{-1}(k)} \right) $$ for all permutations $\sigma$ on $\left\{1, 2, \cdots, k\right\}$ and $$ \nu_{t_1,\cdots,t_k} \left(F_1 \times \cdots F_k \right) = \nu_{t_1,\cdots,t_k,t_{k+1},\cdots,t_{k+m}} \left(F_1 \times \cdots \times F_k \times \mathbb{R}^n \cdots \times \mathbb{R}^n \right) $$ for all $m \in \mathbb{N}$. Then there exists a probability space $(\Omega, \mathcal{F}, P)$ and a stochastic process $\left\{X_t\right\}$ on $\Omega, X_t: \Omega \to \mathbb{R}^n$, such that $$ \nu_{t_1,\cdots,t_k}\left(F_1 \times \cdots \times F_k\right) = P\left[X_{t_1} \in F_1, \cdots, X_{t_k} \in F_k \right], $$ for all $t_i \in T, k \in \mathbb{N}$ and all Borel sets $F_i$.
For what does the notation $\sigma(j)$ and $\sigma^{-1}(j)$ stand for $j = 1, \cdots, k$?
