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).