Composition of a Carathéodory–Lipschitz Function with a Measurable Path

In the setting of Integrals over Subintervals of a Closed Interval: Notation, assume the axiom of dependent choice. Let $\Lambda\in\mathbb{R}$ with $\Lambda\ge0$, let $F\in\mathrm{CL}_{\Lambda}$, the set of Carathéodory–Lipschitz functions with constant $\Lambda$, and for a function $x:[a,b]\to\mathbb{R}^{n}$ let $F(\cdot,x)$ be the function $r\mapsto F(r,x(r))$ on $[a,b]$.

(i) Measurable paths. If $x:[a,b]\to\mathbb{R}^{n}$ is measurable, then $F(\cdot,x)$ is measurable.

(ii) Bounded measurable paths. If $x:[a,b]\to\mathbb{R}^{n}$ is measurable and $M\in\mathbb{R}$ satisfies $\lVert x(r)\rVert\le M$ for every $r\in[a,b]$, then

$$\bigl\lVert F\bigl(r,x(r)\bigr)\bigr\rVert\le\lVert F(r,0)\rVert+\Lambda M\qquad\bigl(r\in[a,b]\bigr),$$

and $F(\cdot,x)\in\mathcal{L}^{1}([a,b];\mathbb{R}^{n})$.

(iii) Continuous paths. If $x:[a,b]\to\mathbb{R}^{n}$ is continuous on $[a,b]$, then $F(\cdot,x)\in\mathcal{L}^{1}([a,b];\mathbb{R}^{n})$.

(iv) Difference bound. For all functions $x,y:[a,b]\to\mathbb{R}^{n}$,

$$\bigl\lVert F\bigl(r,x(r)\bigr)-F\bigl(r,y(r)\bigr)\bigr\rVert\le\Lambda\,\lVert x(r)-y(r)\rVert\qquad\bigl(r\in[a,b]\bigr).$$