Consider a Dirichlet process $P \sim \operatorname{DP}(\alpha)$ on $(X,\mathcal{X})$ and a measurable $\psi : X \mapsto \mathbb{R}$. We know then that:
$$ P \circ \psi^{-1} \sim \operatorname{DP}(\alpha \circ \psi^{-1})$$
However does there exist a relation of this type also for $\int \psi dP$? In other words, can we express the distribution of $\int \psi dP$ in terms of the Dirichlet process as it happens for $P \circ \psi^{-1}$?
The random Dirichlet measure $P$ is purely atomic and is $$P(dw)=\sum_{k=1}^{\infty}\delta_{X_k}(dw)Y_k\prod_{j=1}^{k-1}(1-Y_j)$$ where $X_k$ and $Y_j$ are all independent such that the density of $Y_j$ in $(0,1)$ is $a(1-y)^{a-1}$ where $a$ is the mass of the finite measure $\alpha$ and such that the law on $X_k$ on $\Omega$ is $\alpha/a.$ As a consequence $$\int_{\Omega}\Psi(w)P(dw)=\sum_{k=1}^{\infty}\Psi(X_k)Y_k(\prod_{j=1}^{k-1}(1-Y_j)).$$