Rules for the Riemann Integral of Vector- and Matrix-Valued Maps

In the settings of Single-Variable Calculus on an Interval and Euclidean Space and Real Matrices: Standing Notation, let $I$ be an interval containing two distinct points, as in Single-Variable Calculus on an Interval (ii), let $p,q,m,n\in\mathbb{N}$, and let $x,y:I\to\mathbb{R}^{q}$ and $A,B:I\to\mathcal{M}_{p\times q}(\mathbb{R})$. Integrability on $[u,v]$, the integral $\int_{u}^{v}$ and its being defined are as in Riemann Integral of a Vector-Valued or Matrix-Valued Map (i), Riemann Integral of a Vector-Valued or Matrix-Valued Map (ii), Riemann Integral of a Vector-Valued or Matrix-Valued Map (iii) and Riemann Integral of a Vector-Valued or Matrix-Valued Map (iv), and as in The Riemann Integral on a Closed Interval via Lower and Upper Integrals (viii) for maps $I\to\mathbb{R}$; continuity on $I$ refers to $d_{\mathbb{R}}$ on $I$, $d_{E}$ on $\mathbb{R}^{q}$ and $d_{p\times q}$ on $\mathcal{M}_{p\times q}(\mathbb{R})$.

(i) If $x$ is continuous on $I$, then $x$ is integrable on $[u,v]$ for all $u,v\in I$ with $u<v$, and $\int_{u}^{v}x$ is defined for all $u,v\in I$. The same holds for $A$ in place of $x$.

(ii) Let $u,v\in I$. If $\int_{u}^{v}x$ and $\int_{u}^{v}y$ are defined, then $\int_{u}^{v}\bigl(x(t)+y(t)\bigr)dt$ is defined and equals $\int_{u}^{v}x+\int_{u}^{v}y$. If $\int_{u}^{v}A$ and $\int_{u}^{v}B$ are defined, then $\int_{u}^{v}\bigl(A(t)+B(t)\bigr)dt$ is defined and equals $\int_{u}^{v}A+\int_{u}^{v}B$.

(iii) Let $u,v\in I$ and $c\in\mathbb{R}$. If $\int_{u}^{v}x$ is defined, then $\int_{u}^{v}c\,x(t)\,dt$ is defined and equals $c\int_{u}^{v}x$. If $\int_{u}^{v}A$ is defined, then $\int_{u}^{v}c\,A(t)\,dt$ is defined and equals $c\int_{u}^{v}A$.

(iv) If $x$ is continuous on $I$, then $\int_{u}^{w}x=\int_{u}^{v}x+\int_{v}^{w}x$ for all $u,v,w\in I$. The same holds for $A$ in place of $x$.

(v) Let $b\in\mathbb{R}^{q}$, $C\in\mathcal{M}_{p\times q}(\mathbb{R})$ and $u,v\in I$. The integrals from $u$ to $v$ of the constant maps $I\to\mathbb{R}^{q}$, $t\mapsto b$, and $I\to\mathcal{M}_{p\times q}(\mathbb{R})$, $t\mapsto C$, are defined, and

$$\int_{u}^{v}b\,dt=(v-u)\,b,\qquad\int_{u}^{v}C\,dt=(v-u)\,C .$$

(vi) Let $u,v\in I$, $L\in\mathcal{M}_{m\times q}(\mathbb{R})$, $z\in\mathbb{R}^{q}$, $M\in\mathcal{M}_{m\times p}(\mathbb{R})$ and $N\in\mathcal{M}_{q\times n}(\mathbb{R})$. If $\int_{u}^{v}x$ is defined, then the following integrals are defined and

$$\int_{u}^{v}Lx(t)\,dt=L\int_{u}^{v}x,\qquad\int_{u}^{v}z\cdot x(t)\,dt=z\cdot\int_{u}^{v}x .$$

If $\int_{u}^{v}A$ is defined, then the following integrals are defined and

$$\int_{u}^{v}A(t)z\,dt=\Bigl(\int_{u}^{v}A\Bigr)z,\quad\int_{u}^{v}MA(t)\,dt=M\int_{u}^{v}A,\quad\int_{u}^{v}A(t)N\,dt=\Bigl(\int_{u}^{v}A\Bigr)N,\quad\int_{u}^{v}\bigl(MA(t)\bigr)N\,dt=\Bigl(M\int_{u}^{v}A\Bigr)N .$$

(vii) Let $u,v\in I$. If $\int_{u}^{v}A$ is defined, then $\int_{u}^{v}A(t)^{\top}dt$ is defined and equals $\bigl(\int_{u}^{v}A\bigr)^{\top}$.

(viii) Let $u,v\in I$ with $u\le v$. If $x$ is continuous on $I$, then $\int_{u}^{v}\lVert x(t)\rVert\,dt$ is defined and

$$\Bigl\lVert\int_{u}^{v}x\Bigr\rVert\le\int_{u}^{v}\lVert x(t)\rVert\,dt .$$

If $A$ is continuous on $I$, then $\int_{u}^{v}\lVert A(t)\rVert\,dt$ is defined and $\bigl\lVert\int_{u}^{v}A\bigr\rVert\le\int_{u}^{v}\lVert A(t)\rVert\,dt$, with the operator norm.