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Can a random probability measure be interpreted as random variable with values in the set of probability measures?

Suppose $(\Omega,\mathcal{F},P)$ is a complete probability space and denote by $\mathcal{B}(\mathbb R)$ the Borel $\sigma$-algebra on $\mathbb R$. Recall that a random probability measure is a function $k\colon \Omega\times \mathcal{B}(\mathbb R)\to [0,1]$ such that $E\mapsto k(\omega,E)$ is a probability measure of all $\omega\in\Omega$, $\omega\mapsto k(\omega,E)$ is $\mathcal{F}$-measurable for all $E\in\mathcal{B}(\mathbb R)$. My question is can […]