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 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?