finite rather than merely locally finite.

Note that $(\indfao_{n})_{n=1}^{\infty}$ is not necessarily exchangeable. For instance, consider again . If $\indfao_{1}$ is non-empty, $1 \in \indfao_{1}$ with probability one. If $\indfao_{2}$ is non-empty, with positive probability it may not contain $1$. To restore exchangeability we extend an idea due to @aldous:1985:exchangeability in the setting of random partitions, associating to each feature a draw from a uniform random variable on $[0,1]$. Drawing these random variables independently we maintain consistency across different values of $N$. We refer to these random variables as *uniform random feature labels*.

Note that the use of a uniform distribution is for convenience; we simply require that features receive distinct labels with probability one, so any other continuous distribution would suffice. We also note that in a full-fledged model based on random feature allocations these labels often play the role of parameters and are used in defining the likelihood. For further discussion of such constructions, see @broderick:2012:clusters.

Thus, let $(\randlabfa_{k})$ be a sequence of iid uniform random variables, independent of both $(U_{k})$ and $\randfa_{\infty}$. Construct a new feature labeling by taking the feature labeled $k$ in the order-of-appearance labeling and now label it $\randlabfa_{k}$. In this case, let $\indfar_{n}$ 