Rules for Integrals of Vector- and Matrix-Valued Functions

In the settings of Measurable Functions and the Lebesgue Integral: Notation and Euclidean Space and Real Matrices: Standing Notation, let $(X,\mathcal{F},\mu)$ be a measure space and let $m,n,p,q,r,s\in\mathbb{N}$. Functions into $\mathbb{R}^{n}$ and into $\mathbb{R}^{p\times q}$ are combined pointwise: for functions $f,g:X\to\mathbb{R}^{n}$ and $F,G:X\to\mathbb{R}^{p\times q}$, $a,b\in\mathbb{R}$, a point $u\in\mathbb{R}^{n}$ and matrices $C$, $D$ of matching sizes, $af+bg$, $aF+bG$, $Cf$, $CFD$ and $u\cdot f$, with the dot product $u\cdot f(x)$, are the functions

$$x\mapsto af(x)+bg(x),\quad x\mapsto aF(x)+bG(x),\quad x\mapsto Cf(x),\quad x\mapsto\bigl(CF(x)\bigr)D,\quad x\mapsto u\cdot f(x),$$

and $\lVert f\rVert$ and $\lVert F\rVert$ are the functions $x\mapsto\lVert f(x)\rVert$ and $x\mapsto\lVert F(x)\rVert$ from $X$ to $\mathbb{R}$, with the Euclidean norm and the operator norm (these functions exist by Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (i), Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (iii) and Maps Formed Pointwise from Vector and Matrix Operations (i)). Integrals of $\lVert f\rVert$ and $\lVert F\rVert$ are taken in $\mathcal{L}^{1}(\mu)$.

(i) Integrability criterion for vector-valued functions. Let $f:X\to\mathbb{R}^{n}$ be measurable. Then $f\in\mathcal{L}^{1}(\mu;\mathbb{R}^{n})$ if and only if $\lVert f\rVert\in\mathcal{L}^{1}(\mu)$.

(ii) Integrability criterion for matrix-valued functions. Let $F:X\to\mathbb{R}^{p\times q}$ be measurable. Then $F\in\mathcal{L}^{1}(\mu;\mathbb{R}^{p\times q})$ if and only if $\lVert F\rVert\in\mathcal{L}^{1}(\mu)$.

(iii) Linearity for vector-valued functions. Let $f,g\in\mathcal{L}^{1}(\mu;\mathbb{R}^{n})$ and $a,b\in\mathbb{R}$. Then $af+bg\in\mathcal{L}^{1}(\mu;\mathbb{R}^{n})$ and

$$\int_{X}(af+bg)\,d\mu=a\int_{X}f\,d\mu+b\int_{X}g\,d\mu .$$

(iv) Linearity for matrix-valued functions. Let $F,G\in\mathcal{L}^{1}(\mu;\mathbb{R}^{p\times q})$ and $a,b\in\mathbb{R}$. Then $aF+bG\in\mathcal{L}^{1}(\mu;\mathbb{R}^{p\times q})$ and

$$\int_{X}(aF+bG)\,d\mu=a\int_{X}F\,d\mu+b\int_{X}G\,d\mu .$$

(v) Constant matrix factors of vector-valued functions. Let $C\in\mathcal{M}_{m\times n}(\mathbb{R})$ and $f\in\mathcal{L}^{1}(\mu;\mathbb{R}^{n})$. Then $Cf\in\mathcal{L}^{1}(\mu;\mathbb{R}^{m})$ and

$$\int_{X}Cf\,d\mu=C\int_{X}f\,d\mu .$$

(vi) Constant matrix factors of matrix-valued functions. Let $C\in\mathcal{M}_{r\times p}(\mathbb{R})$, $D\in\mathcal{M}_{q\times s}(\mathbb{R})$ and $F\in\mathcal{L}^{1}(\mu;\mathbb{R}^{p\times q})$. Then $CFD\in\mathcal{L}^{1}(\mu;\mathbb{R}^{r\times s})$ and

$$\int_{X}CFD\,d\mu=\Bigl(C\int_{X}F\,d\mu\Bigr)D .$$

(vii) Dot products. Let $u\in\mathbb{R}^{n}$ and $f\in\mathcal{L}^{1}(\mu;\mathbb{R}^{n})$. Then $u\cdot f\in\mathcal{L}^{1}(\mu)$ and

$$\int_{X}u\cdot f\,d\mu=u\cdot\int_{X}f\,d\mu .$$

(viii) Norm bound for vector-valued functions. Let $f\in\mathcal{L}^{1}(\mu;\mathbb{R}^{n})$. Then $\lVert f\rVert\in\mathcal{L}^{1}(\mu)$ and

$$\Bigl\lVert\int_{X}f\,d\mu\Bigr\rVert\le\int_{X}\lVert f\rVert\,d\mu .$$

(ix) Norm bound for matrix-valued functions. Let $F\in\mathcal{L}^{1}(\mu;\mathbb{R}^{p\times q})$. Then $\lVert F\rVert\in\mathcal{L}^{1}(\mu)$ and

$$\Bigl\lVert\int_{X}F\,d\mu\Bigr\rVert\le\int_{X}\lVert F\rVert\,d\mu .$$