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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?