Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and let $\mathcal{M}(\mathbb{R}^d)$ be the space of finite signed measures on $\mathbb{R}^d$ endowed with the narrow topology (i.e. the initial topology w.r.t. $C_b(\mathbb{R}^d)$, the set of real valued, continuous and bounded functions on $\mathbb{R}^d$) and the corresponding Borel $\sigma$-algebra. Let $\mu: \Omega \to \mathcal{M}(\mathbb{R}^d)$ be measurable and let us define the multifunction $$F : \Omega \rightrightarrows C_{0,1}(\mathbb{R}^d):=\{ \varphi \in C_0(\mathbb{R}^d) \mid |\varphi|_{\infty} \le 1 \}$$ (where $C_0(\mathbb{R}^d)$ is the Banach space of continuous functions vanishing at infinity with the supremum norm) as $$F(\omega) := \left \{ \varphi \in C_{0,1}(\mathbb{R}^d) \mid \int_{\mathbb{R}^d} \varphi \text{ d} \mu_{\omega} \ge \frac{|\mu_{\omega}|}{2} \right \}, $$ where $|\mu_{\omega}|$ is the total variation norm of $\mu_{\omega}$.

Can we find a measurable selection $f: \Omega \to C_{0,1}(\mathbb{R}^d)$ of $F$, meaning that $f$ is measurable and $f(\omega) \in F(\omega)$ for every $\omega \in \Omega$?

I tried with the Kuratowski–Ryll-Nardzewski measurable selection theorem but I am not able to prove that $\{ \omega \in \Omega \mid F(\omega) \cap U \}$ is measurable for every $U \subset C_{0,1}(\mathbb{R}^d)$ open.

Any hint would be really appreciated!

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