Mean measure of Poisson-Dirichlet components (${\alpha }_{\theta }(dx)=\theta x^{-1}(1-x{)}^{\theta -1}dx$)

For $PD(\theta )$, $\mathbb{E}{\sum }_i\phi (p_i)=\int \phi d{\alpha }_{\theta }$ whenever the absolute integral is finite. Apply the Mecke formula for a Poisson random measure to a selected jump $u$ and the remaining total $t$, whose density is $t^{\theta -1}{e}^{-t}/\Gamma (\theta )$. Changing variables to $s=u+t$ and $x=u/s$ factors out a Gamma function and gives the displayed measure. Weighting it by $x$ gives a probability density $\theta (1-x{)}^{\theta -1}$, the Beta distribution of a size-biased component.