Functions and Function-Valued Operators Given by Integrals over Subintervals

In the setting of Integrals over Subintervals of a Closed Interval: Notation, assume the axiom of dependent choice. For sets $Y$ and $Z$, $Y^{Z}$ is the set of functions from $Z$ to $Y$. For a function $h:[a,b]\times[a,b]\to\mathbb{R}^{n}$ and $t\in[a,b]$, $h(t,\cdot):[a,b]\to\mathbb{R}^{n}$ is the function $s\mapsto h(t,s)$ of Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (iv).

(i) Scalar integrands. For every $f\in\mathcal{L}^{1}([a,b])$ there is exactly one function $X:[a,b]\to\mathbb{R}$ with

$$X(t)=\int_{[a,t]}f\,d\lambda\qquad(t\in[a,b]).$$

(ii) Vector-valued integrands. For every $g\in\mathcal{L}^{1}([a,b];\mathbb{R}^{n})$ there is exactly one function $X:[a,b]\to\mathbb{R}^{n}$ with

$$X(t)=\int_{[a,t]}g\,d\lambda\qquad(t\in[a,b]).$$

(iii) The integral operator. There is exactly one function $\mathcal{J}:\mathcal{L}^{1}([a,b];\mathbb{R}^{n})\to(\mathbb{R}^{n})^{[a,b]}$ with

$$\mathcal{J}(g)(t)=\int_{[a,t]}g\,d\lambda\qquad(g\in\mathcal{L}^{1}([a,b];\mathbb{R}^{n}),\ t\in[a,b]).$$

(iv) The substitution operator. For every function $F:[a,b]\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ there is exactly one function $\Theta:(\mathbb{R}^{n})^{[a,b]}\to(\mathbb{R}^{n})^{[a,b]}$ with

$$\Theta(z)(r)=F(r,z(r))\qquad(z\in(\mathbb{R}^{n})^{[a,b]},\ r\in[a,b]).$$

(v) Integrands depending on the endpoint. Let $h:[a,b]\times[a,b]\to\mathbb{R}^{n}$ be a function with $h(t,\cdot)\in\mathcal{L}^{1}([a,b];\mathbb{R}^{n})$ for every $t\in[a,b]$. There is exactly one function $Y:[a,b]\to\mathbb{R}^{n}$ with

$$Y(t)=\int_{[a,t]}h(t,s)\,ds\qquad(t\in[a,b]).$$