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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

  1. $E\mapsto k(\omega,E)$ is a probability measure of all $\omega\in\Omega$,

  2. $\omega\mapsto k(\omega,E)$ is $\mathcal{F}$-measurable for all $E\in\mathcal{B}(\mathbb R)$.

My question is can a random probability measure be interpreted as random variable with values in the set of probability measures?

More preciselly:

Consider the space $\mathcal{P}$ of all Borel probability measures on $\mathbb R$ which we endow with the weak topology.
Consider the Borel $\sigma$-algebra $\mathcal{B}(\mathcal{P})$.

Suppose that $k\colon\Omega\to \mathcal{P}$ is $\mathcal{F}$$\mathcal{B}(\mathcal{P})$-measurable, is it true that $(\omega,E)\mapsto k(\omega)(E)$ is a random Borel probaility?

Conversely, if $k\colon \Omega\times \mathcal{B}(\mathbb R)\to [0,1]$ is a random Borel probability, is it true that the map $\omega\mapsto k(\omega,\cdot)$ is $\mathcal{F}$$\mathcal{B}(\mathcal{P})$-measurable?

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