Dirichlet process

Let $(\mathsf{X},\mathcal{A})$ be a standard Borel space and let $\alpha$ be a finite nonzero measure on it. Write $\mathcal{P}(\mathsf{X})$ for the set of probability measures on $(\mathsf{X},\mathcal{A})$, equipped with the evaluation sigma-algebra $\sigma (\{G\mapsto G(A):A\in \mathcal{A}\})\mathrm{.}$ For a probability law $P$ on this space, let $G$ be the coordinate random probability measure $G(\omega )=\omega$. Call $P$ a Dirichlet-process law with base measure $\alpha$ if, for every finite measurable partition $(A_1,\dots ,A_k)$ of $\mathsf{X}$, the vector $(G(A_1),\dots ,G(A_k))$ has the Dirichlet distribution with parameters $(\alpha (A_1),\dots ,\alpha (A_k))$ after zero-parameter cells are omitted; each omitted zero-base cell has mass zero. If the partition has one positive-parameter cell, its mass is one. The finite-partition specification determines at most one law on the evaluation sigma-algebra. When such a law exists, denote it by $DP(\alpha )$; the construction theorem on this page proves existence.

The Dirichlet distribution here has the positive-parameter density and one-category convention of Dirichlet distribution, including the one-category case. The definition asserts no weak topology, support statement, or density-contraction result.