One marked-sum law determines random mass partitions and $Ξ$-coalescents

Let $P=(P_j)$ be a random mass partition and let $(X_j)$ be iid real marks, independent of $P$. We construct a fixed mark distribution for which the law of the real random variable $\sum_j P_j X_j$ determines the law of $P$ on the Kingman simplex. The observation map is an affine topological embeddi