Let $\mathcal{P}_2(\mathbb{R})$ be the space of measures on $\mathbb{R}$ with finite second moment. We equip this space with the Wasserstein metric $W^2.$ We recall tat this metric space is separable and complete.
Let $b: [0,T] \times \Omega \times \mathbb{R} \times \mathcal{P}_2(\mathbb{R}) \rightarrow \mathbb{R}$ with the properties:
For all $ x \in \mathbb{R}, \mu \in \mathcal{P}_2(\mathbb{R})$ the process $(t,\omega) \mapsto b(t,\omega, x, \mu)$ is progressively measurable.
There exits $ C > 0 $ such that for all $\omega, t, x_1, x_2, \mu_1, \mu_2 $ we have $$ | b(t,\omega, x_1,\mu_1) - b(t,\omega, x_2,\mu_2) | \leq C \big( | x_1 - x_2 | + W^2(\mu_1, \mu_2) \big) . $$
Let $X = (X_t)_{t \in [0,T]}$ be progressively measurable. Denote by $ P(X_t)$ the distribution of $X_t.$ I want to show that the process $$ (t, \omega) \mapsto b(t,\omega, X_t(\omega), P(X_t) ) $$ is progressively measurable. This should follow from 1. and 2. but I was not able to prove it. I hope someone can help me out.