Ordinary Differential Equations with Measurable Time Dependence: Standing Notation and Background

(i) Base. This setting extends Integrals over Subintervals of a Closed Interval: Notation: its conventions and notation, and through it those of Ordinary Differential Equations on a Compact Interval: Standing Notation and Measurable Functions and the Lebesgue Integral: Standing Notation, are in force.

(ii) Integrals over subintervals. Notation as in Integrals over Subintervals of a Closed Interval: Notation (ii) and Integrals over Subintervals of a Closed Interval: Notation (iii). Background: Absolute Continuity of the Integral, Indefinite Lebesgue Integrals on a Closed Interval and Gronwall's Lemma for Bounded Measurable Functions; for the formal bookkeeping: Functions and Function-Valued Operators Given by Integrals over Subintervals.

(iii) Carathéodory solutions. For a function $F:[a,b]\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ and $\xi\in\mathbb{R}^{n}$, the integral equation is that of The Integral Equation of an Initial Value Problem with Measurable Time Dependence (i), and its Carathéodory solutions and the set $\mathrm{Sol}(F,\xi)$ of them are as in Carathéodory Solutions of an Initial Value Problem (i) and Carathéodory Solutions of an Initial Value Problem (ii). For a real number $\Lambda\ge0$, $\mathrm{CL}_{\Lambda}$ is the set of Carathéodory–Lipschitz functions with constant $\Lambda$. Background: Composition of a Carathéodory–Lipschitz Function with a Measurable Path, Existence and Uniqueness for Ordinary Differential Equations with Measurable Time Dependence and Variation of Constants with Integrable Forcing.