Let $(\mathcal{X}, \mathcal{B})$ be a measurable space and let its probability measure be $P$. In Bayesian statistics, we may wish to define a prior $\mu$ on the space of all such probability measures, say $\mathcal{P}$. Can we say $\mu$ is a random probability measure on $\mathcal{X}$?
For instance, the $k$-dimensional Dirichlet distribution defines a probability measure over a $(k-1)$-dimensional probability simplex $\Delta_k = \{q \in \mathbb{R}^k | \sum_{i=1}^k q_i = 1, q_i \geq 0, i = 1, \ldots, k\}$, which is a set of all probability measures on a finite set $\mathcal{X} = \{1, \ldots, k\}$. In that sense, is $\mu$ a random probability measure on $\mathcal{X}$?