Exchangeable Probability Distribution

Let {⚠ $Z$} be a measurable space. A probability distribution {⚠ $P$} on the measurable space {⚠ $Z^n$} of sequences of length {⚠ $n$}, where {⚠ $n \in {1, 2, \ldots}$}, is exchangeable if

{⚠ $P(E) = P\{z_1, \ldots, z_n: z_{\pi(1)}, \ldots, z_{\pi(n)}\in E\}$}

for any measurable {⚠ $E \subseteq Z^n$} and any permutation {⚠ $\pi$} of the set {⚠ $\{1, \ldots, n\}$} (i.e., if the distribution of the sequence {⚠ $z_1, \ldots, z_n$} is invariant under any permutation of the indices).

A probability distribution {⚠ $P$} on the power measurable space {⚠ $Z^{\infty}$} is exchangeable if the marginal distribution {⚠ $P_n$} of {⚠ $P$} on {⚠ $Z^n$} (defined by

{⚠ $P_n(E) := P\{(z_1, z_2, \ldots) \in Z^{\infty}: z_1, \ldots, z_n\in E\}$}

for all events {⚠ $E \subseteq Z^n$}) is exchangeable for each {⚠ $n = 1, 2, \ldots$} (i.e., if the distribution of the sequence {⚠ $z_1, z_2, \ldots$} is invariant under any permutation of the finite number of the indices).