Definition 14.10.1 (Consistent). Let $\bracs{(\Omega_i, \cf_i)}_{i \in I}$ be a family of measurable spaces. For each $J \subset J' \subset I$, denote $\pi_{J}: \prod_{i \in I}X_{i} \to \prod_{j \in J}X_{j}$ and $\pi_{J', J}: \prod_{j \in J'}X_{j} \to \prod_{j \in J}X_{j}$ as the projection maps.

Let $\bracs{\mu_J|J \subset I \text{ finite}}$ such that each $\mu_{J}$ is a probability measure on $\prod_{j \in J}\Omega_{j}$, then $\bracs{\mu_J}$ is consistent if for all $J \subset J' \subset I$, $\mu_{J'}= \mu_{J} \circ \pi_{J', J}^{-1}$.