A sequence \((X_n)_{n \in \mathbb {N}}\) of random variables is exchangeable with respect to measure \(\mu \) if for every \(n \in \mathbb {N}\) and every permutation \(\sigma \) of \(\{ 0, \ldots , n-1\} \), the joint distribution of \((X_{\sigma (0)}, \ldots , X_{\sigma (n-1)})\) equals that of \((X_0, \ldots , X_{n-1})\).

A sequence \((X_n)_{n \in \mathbb {N}}\) is contractable with respect to measure \(\mu \) if for all \(m \in \mathbb {N}\) and all strictly increasing functions \(k, k' : \mathrm{Fin}(m) \to \mathbb {N}\), the joint distribution of \((X_{k(0)}, \ldots , X_{k(m-1)})\) equals that of \((X_{k'(0)}, \ldots , X_{k'(m-1)})\).

A sequence \((X_n)_{n \in \mathbb {N}}\) is conditionally i.i.d. with respect to measure \(\mu \) if there exists a probability kernel \(\nu : \Omega \to \mathrm{Measure}(\alpha )\) such that for any strictly monotone \(k : \mathrm{Fin}(m) \to \mathbb {N}\), the joint distribution of \((X_{k(0)}, \ldots , X_{k(m-1)})\) equals the mixture \(\mu .\mathrm{bind}(\omega \mapsto \nu (\omega )^{\otimes m})\).

The tail \(\sigma \)-algebra of a sequence \((X_n)\) is \(\mathcal{T} = \bigcap _{n=0}^{\infty } \sigma (X_n, X_{n+1}, \ldots )\).

The future filtration at level \(m\) is \(\mathcal{F}_m = \sigma (X_{m+1}, X_{m+2}, \ldots )\).

The shift-invariant \(\sigma \)-algebra consists of sets \(S\) such that \(\theta ^{-1}(S) = S\) where \(\theta \) is the shift operator.