I have seen the following two definitions for random probability measures:
Suppose that $(\Omega,\mathcal F,P)$ is a (possibly complete) probability space and $B(\mathbb R)$ denote the Borel sigma-algebra on $\mathbb R$.
Definition 1: A function $k\colon \Omega\times \mathcal B(\mathbb R)\to [0,1]$ is called a random probability measure if $\omega\mapsto k(\omega,E)$ is $\mathcal F$-measurable for all $E\in\mathcal B(\mathbb R)$, and $E\mapsto k(\omega,E)$ is a probability measure for all $\omega$.
Definition 2: Denote by $\mathcal P(\mathbb R)$ the set of all Borel probability measures. A function $k\colon \Omega\to \mathcal P(\mathbb R)$ is called a random probability measure if $k$ is $\mathcal F$-measurable, where $\mathcal P(\mathbb R)$ is endowed with the Borel sigma-algebra $\mathcal B(\mathcal P(\mathbb R))$.
My question is: Are both definitions equivalent?