Carathéodory–Lipschitz Right-Hand Sides

In the setting of Integrals over Subintervals of a Closed Interval: Notation, assume the axiom of dependent choice. Let $F:[a,b]\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ be a function and let $\Lambda\in\mathbb{R}$ with $\Lambda\ge0$. $F$ is Carathéodory–Lipschitz with constant $\Lambda$ if the following three conditions hold.

(i) Measurability in time. For every $y\in\mathbb{R}^{n}$, the function $r\mapsto F(r,y)$ from $[a,b]$ to $\mathbb{R}^{n}$ is measurable with respect to $\mathcal{B}_{[a,b]}$.

(ii) Lipschitz dependence on the state.

$$\lVert F(r,y)-F(r,z)\rVert\le\Lambda\,\lVert y-z\rVert\qquad\bigl(r\in[a,b],\ y,z\in\mathbb{R}^{n}\bigr).$$

(iii) Integrability at the origin. The function $r\mapsto F(r,0)$ on $[a,b]$ lies in $\mathcal{L}^{1}([a,b];\mathbb{R}^{n})$.

(iv) The class $\mathrm{CL}_{\Lambda}$. $\mathrm{CL}_{\Lambda}$ is the set of those elements of the set $(\mathbb{R}^{n})^{[a,b]\times\mathbb{R}^{n}}$ of functions from $[a,b]\times\mathbb{R}^{n}$ to $\mathbb{R}^{n}$ that are Carathéodory–Lipschitz with constant $\Lambda$ (a subset of a set by The Elements of the Class of Functions from a Set to a Set, and This Class Is a Set (ii), the condition being predicative, with the Lipschitz condition written through the Euclidean distance and the real operations of Arithmetic and Order Rules in the Real Numbers, Stated for ℝ (i); by The Elements of the Class of Functions from a Set to a Set, and This Class Is a Set (i), $F\in\mathrm{CL}_{\Lambda}$ if and only if $F$ is a function $[a,b]\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ that is Carathéodory–Lipschitz with constant $\Lambda$).