For a Krasovskii Regularized System, does there exist a Cartheodory Solution for all $t_{0},x_{0}\in [0,\infty)\times\mathbb{R}^{d}$ w/ IC x(t_0)=x_0

Question

The differential inclusion associated with a time-varying set valued map $F:[0,\infty)\times\mathbb{R}^{d}\to\mathfrak{B}(\mathbb{R}^d)$ is an equation of the following form $$\dot{x}(t)\in F(t,x(t)).$$

Here $\mathfrak{B}(S)$ is the set whose elements are all of the possible subsets of $S\in\mathbb{R}^d$.

Following proposition S2 of Discontinuous Dynamical Systems: A Tutorial on Solutions, Nonsmooth Analysis, and Stability by Jorge Cortes:

Let $F:[0,\infty)\times\mathbb{R}^{d}\to\mathfrak{B}(\mathbb{R}^d)$ be locally bounded and take non-empty, compact, and convex values. Assume that, for each $t\in\mathbb{R}$, the set-value map $x\mapsto F(t,x)$ is upper semicontinuous, and, for each $x\in\mathbb{R}^{d}$, the set-valued map $t\mapsto F(t,x)$ is measurable. Then for all $(t_{0},x_{0})\in[0,\infty)\times\mathbb{R}^{d}$, there exists a Caratheodory solution with initial condition (IC) $x(t_{0})=x_0$.

Next, the Krasovskii regularization for $F$ is defined as $$\hat{F}(x)\triangleq\bigcap_{\delta >0}\overline{\text{co}}F(x+\delta\mathbb{B}).$$

Note, I am taking this definition from Hybrid Dynamical Systems: Modeling, Stability, and Robustness by Goebel, Sanfelice, and Teel.

To me it seems that Krasovskii regularization satisfies all the requirements for the existence of Caratheodory solutions, but I am uncertain about the measurability condition. I am not even sure what it means for a set-valued mapping to be measurable.

Additional Information:

Example 6.6 and Lemma 5.16 of the Goebel text gets me the following:

1. $\hat{F}$ is outer semicontinuous
2. $\hat{F}$ is locally bounded relative to a flow set $C$
3. $\hat{F}$ is convex valued for every $x\in C$

For a flow map $F$ that evolves over a flow set $C$, the Krasovskii regularization is $$\hat{F}(x)\triangleq\bigcap_{\delta >0}\overline{\text{co}}F((x+\delta\mathbb{B})\cap C),\quad\forall x\in C.$$

If additional information is needed, let me know, I'll do my best to fix the problem.

Upper semicontinuity information:

Image of text: Set Valued maps information Link to text: https://bonetto.math.gatech.edu/teaching/6307-fall20/DiscDynSys.pdf

See page 74 and 75

In Aubin's Set-Valued Analysis text I found the following on measurability: Measurable Set-Valued Maps

From the same text: If $F$ is upper semi-continuous (or lower), then $F$ is measurable. Continuity implies measurability