The Integral Equation of an Initial Value Problem with Measurable Time Dependence

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 $\xi\in\mathbb{R}^{n}$.

(i) The integral equation. For an unknown function $x:[a,b]\to\mathbb{R}^{n}$, the integral equation on $[a,b]$ with right-hand side $F$ and initial value $\xi$ is

$$x(t)=\xi+\int_{[a,t]}F\bigl(r,x(r)\bigr)\,dr\qquad\bigl(t\in[a,b]\bigr).$$