Integrals over Subintervals of a Closed Interval: Notation

(i) Base. This setting extends Ordinary Differential Equations on a Compact Interval: Standing Notation and Measurable Functions and the Lebesgue Integral: Standing Notation: their conventions and notation are in force. Below, $a,b\in\mathbb{R}$ with $a<b$, $n\in\mathbb{N}$, and $([a,b],\mathcal{B}_{[a,b]},\lambda_{[a,b]})$ is the restriction of Lebesgue measure to $[a,b]$, as in Measurable Functions and the Lebesgue Integral: Standing Notation (vii).

(ii) Integrable functions on $[a,b]$. $\mathcal{L}^{1}([a,b])$ and $\mathcal{L}^{1}([a,b];\mathbb{R}^{n})$ denote $\mathcal{L}^{1}(\lambda_{[a,b]})$ and $\mathcal{L}^{1}(\lambda_{[a,b]};\mathbb{R}^{n})$.

(iii) Integrals over subintervals. Let $a\le s\le t\le b$, and let $([s,t],\mathcal{B}_{[s,t]},\lambda_{[s,t]})$ be the restriction of $([a,b],\mathcal{B}_{[a,b]},\lambda_{[a,b]})$ to $[s,t]$ (here $[s,t]\in\mathcal{B}_{[a,b]}$, since $[s,t]\subseteq[a,b]$ and intervals are Borel sets by Intervals Are Borel, and Rays, Half-Open Intervals and Rational Intervals Generate the Borel Sigma-Algebra of the Real Line (ii)). For a function $f:[a,b]\to\mathbb{R}$ with $f|_{[s,t]}\in\mathcal{L}^{1}(\lambda_{[s,t]})$, and for a function $g:[a,b]\to\mathbb{R}^{n}$ with $g|_{[s,t]}\in\mathcal{L}^{1}(\lambda_{[s,t]};\mathbb{R}^{n})$,

$$\int_{[s,t]}f\,d\lambda=\int f|_{[s,t]}\,d\lambda_{[s,t]}\in\mathbb{R},\qquad \int_{[s,t]}g\,d\lambda=\int g|_{[s,t]}\,d\lambda_{[s,t]}\in\mathbb{R}^{n}.$$

When the integrand is given by an expression $h(r)$ in $r\in[a,b]$, $\int_{[s,t]}h(r)\,dr$ denotes $\int_{[s,t]}h\,d\lambda$ for the function $h:r\mapsto h(r)$.