Proof of Rules for Integrals of Vector- and Matrix-Valued Functions

Conventions. For $N,K\in\mathbb{N}$, $g_{k}$ ($k\in[N]$) are the components of a function $g:X\to\mathbb{R}^{N}$ and $G_{ij}$ ($i\in[N]$, $j\in[K]$) the entries of a function $G:X\to\mathbb{R}^{N\times K}$, as in Measurable Functions and the Lebesgue Integral: Notation (x); so $g_{k}(x)=g(x)_{k}$ and $G_{ij}(x)=G(x)_{ij}$ (Component Maps and Entry Maps (i), Component Maps and Entry Maps (ii)). Elements of $\mathcal{L}^{1}(\mu)$ are measurable functions $X\to\mathbb{R}$, and with $g\in\mathcal{L}^{1}(\mu)$ also $|g|\in\mathcal{L}^{1}(\mu)$ (Integrable Functions and Their Lebesgue Integral (ii), Positive Part, Negative Part and Absolute Value of a Real-Valued Function (iii), Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (x)). Arithmetic and order of real numbers, such as $|y|=y$ for $y\ge0$ and $y\le|y|$, are those of the ordered field $\mathbb{R}$, also read in the type of real numbers (The Real Numbers Form an Ordered Field in Which Every Nonempty Set Bounded Above Has a Supremum (i), Rules of Arithmetic and Order in an Ordered Field (x), The Real Numbers as a Type (iv), The Real Numbers as a Type (v), The Real Numbers as a Type (vi), Differences, Quotients and Absolute Values of Real Numbers, and Their Order in the Extended Real Line, Read in the Type Real (iii)). Every sum $\sum_{j=1}^{N}$ below, whether an iterated sum or a sum over the interval $\{1,\dots,N\}$ as in dot products and matrix products, is the sum over the finite set $[N]$, and for every function $\Phi$ on $[N]\times X$ the function $x\mapsto\sum_{j=1}^{N}\Phi(j,x)$ exists (Finite Sets: the Pigeonhole Principle, Uniqueness of the Length, Subsets, Unions, Products, Images, Bounded Sets of Natural Numbers, Extreme Elements, Sets of Maps, Finite Unions and Finite Choice (vii), Natural Numbers as Indices: ℕ ⊆ ℕ₀, 1 ∈ ℕ₀, and the Two Readings of an Operation from 1 to n (iii), Natural Numbers as Indices: ℕ ⊆ ℕ₀, 1 ∈ ℕ₀, and the Two Readings of an Operation from 1 to n (iv), Maps Given by Iterated Operations: Partial Iterates, Images of Natural Numbers, Powers and Operations over a Finite Set (v)).

Step 1: termwise integration. Let $N\in\mathbb{N}$, let $\Phi:[N]\times X\to\mathbb{R}$ have $\Phi(j,\cdot)\in\mathcal{L}^{1}(\mu)$ for every $j\in[N]$, and let $c:[N]\to\mathbb{R}$. By Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (viii), the function $x\mapsto\sum_{j=1}^{N}c_{j}\Phi(j,x)$ lies in $\mathcal{L}^{1}(\mu)$ and

$$\int_{X}\sum_{j=1}^{N}c_{j}\Phi(j,\cdot)\,d\mu=\sum_{j=1}^{N}c_{j}\int_{X}\Phi(j,\cdot)\,d\mu .$$

This is applied to $\Phi(k,x)=g(x)_{k}$ and $\Phi(k,x)=|g(x)_{k}|$ for $g\in\mathcal{L}^{1}(\mu;\mathbb{R}^{N})$, whose sections $\Phi(k,\cdot)$ are $g_{k}$ and $|g_{k}|$, and to $\Phi(b,x)=G(x)_{ib}$, $\Phi(a,x)=G(x)_{aj}$ and $\Phi(b,x)=|G(x)_{ib}|$ for $G\in\mathcal{L}^{1}(\mu;\mathbb{R}^{N\times K})$, whose sections are $G_{ib}$, $G_{aj}$ and $|G_{ib}|$; all these sections lie in $\mathcal{L}^{1}(\mu)$ by Integrable Vector- and Matrix-Valued Functions and Their Integrals (i), Integrable Vector- and Matrix-Valued Functions and Their Integrals (iii) and the conventions (the functions $\Phi$ exist and have these sections by Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (i), Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (i), Projection, Evaluation and Currying Maps (i), Projection, Evaluation and Currying Maps (ii), 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 (ii), Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (iii), Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (iv), Absolute Value and Squaring as Maps on the Reals, and Entrywise Absolute Values and Squares of a Real Matrix (i) and Values, Restriction and Equality of a Map, and the Map $u\mapsto t(u)$ as a Defined Set Symbol (iv)).

Step 2: assembly from components. Let $N,K\in\mathbb{N}$. If $g:X\to\mathbb{R}^{N}$ has $g_{k}\in\mathcal{L}^{1}(\mu)$ for every $k\in[N]$, then the $g_{k}$ are measurable, so $g$ is measurable, $g\in\mathcal{L}^{1}(\mu;\mathbb{R}^{N})$, and

$$\Bigl(\int_{X}g\,d\mu\Bigr)_{k}=\int_{X}g_{k}\,d\mu\qquad(k\in[N])$$

(Vector- and Matrix-Valued Measurable Maps: Components, Entries and Continuous Images (i), Integrable Vector- and Matrix-Valued Functions and Their Integrals (i), Integrable Vector- and Matrix-Valued Functions and Their Integrals (ii)). Likewise, if $G:X\to\mathbb{R}^{N\times K}$ has $G_{ij}\in\mathcal{L}^{1}(\mu)$ for all $i,j$, then $G\in\mathcal{L}^{1}(\mu;\mathbb{R}^{N\times K})$ and $\bigl(\int_{X}G\,d\mu\bigr)_{ij}=\int_{X}G_{ij}\,d\mu$ (Vector- and Matrix-Valued Measurable Maps: Components, Entries and Continuous Images (ii), Integrable Vector- and Matrix-Valued Functions and Their Integrals (iii), Integrable Vector- and Matrix-Valued Functions and Their Integrals (iv)). Two points of $\mathbb{R}^{N}$, or two matrices of the same size, with the same components are equal (Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (iii), Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (iii)).

Proof of (i). The function $\lVert f\rVert$ is measurable (Sums, Products, Transposes, Dot Products and Norms of Measurable Vector- and Matrix-Valued Maps (iii)). For $x\in X$ and $k\in[n]$, $\lVert f(x)\rVert\ge0$, so $\bigl|\lVert f(x)\rVert\bigr|=\lVert f(x)\rVert$, and

$$|f_{k}(x)|\le\lVert f(x)\rVert\le\sum_{k'=1}^{n}|f_{k'}(x)|$$

(Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (i), Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (iv)). If $f\in\mathcal{L}^{1}(\mu;\mathbb{R}^{n})$, the right-hand side is a function in $\mathcal{L}^{1}(\mu)$ by Step 1 with $c_{k}=1$, and so $\lVert f\rVert\in\mathcal{L}^{1}(\mu)$ (Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (v)). If $\lVert f\rVert\in\mathcal{L}^{1}(\mu)$, each $f_{k}$ is measurable and dominated by $\lVert f\rVert$, so $f_{k}\in\mathcal{L}^{1}(\mu)$, and $f\in\mathcal{L}^{1}(\mu;\mathbb{R}^{n})$ (Vector- and Matrix-Valued Measurable Maps: Components, Entries and Continuous Images (i), Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (v), Integrable Vector- and Matrix-Valued Functions and Their Integrals (i)).

Proof of (ii). The function $\lVert F\rVert$ is measurable (Sums, Products, Transposes, Dot Products and Norms of Measurable Vector- and Matrix-Valued Maps (iii)). For $x\in X$, $i\in[p]$ and $j\in[q]$, $\lVert F(x)\rVert\ge0$ and

$$|F_{ij}(x)|\le\lVert F(x)\rVert\le\sum_{k=1}^{p}\sum_{l=1}^{q}|F_{kl}(x)|$$

(Rules for the Operator Norm: Bound, Least Constant, Definiteness, Homogeneity, Triangle Inequality and Products (i), The Operator Norm of Entries, Transposes, Multiples of the Identity, Orthogonal Factors and Diagonal Matrices (ii), The Operator Norm of Entries, Transposes, Multiples of the Identity, Orthogonal Factors and Diagonal Matrices (iv)). If $F\in\mathcal{L}^{1}(\mu;\mathbb{R}^{p\times q})$, Step 1 with $c_{l}=1$ gives $\rho_{k}=\sum_{l=1}^{q}|F_{kl}|\in\mathcal{L}^{1}(\mu)$ for every $k\in[p]$, and Step 1 again, with $c_{k}=1$ and $\Psi(k,x)=\rho_{k}(x)$, gives that the right-hand side is a function in $\mathcal{L}^{1}(\mu)$ (the function $\Psi$ on $[p]\times X$ exists by Maps Given by Iterated Operations: Partial Iterates, Images of Natural Numbers, Powers and Operations over a Finite Set (v), and $\Psi(k,\cdot)=\rho_{k}$ by Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (iv) and Values, Restriction and Equality of a Map, and the Map $u\mapsto t(u)$ as a Defined Set Symbol (iv)); so $\lVert F\rVert\in\mathcal{L}^{1}(\mu)$ (Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (v)). If $\lVert F\rVert\in\mathcal{L}^{1}(\mu)$, each $F_{ij}$ is measurable and dominated by $\lVert F\rVert$, so $F_{ij}\in\mathcal{L}^{1}(\mu)$, and $F\in\mathcal{L}^{1}(\mu;\mathbb{R}^{p\times q})$ (Vector- and Matrix-Valued Measurable Maps: Components, Entries and Continuous Images (ii), Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (v), Integrable Vector- and Matrix-Valued Functions and Their Integrals (iii)).

Proof of (iii). Let $h=af+bg$. For $k\in[n]$, $h_{k}(x)=af_{k}(x)+bg_{k}(x)$ for every $x\in X$ (Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (iv)), so $h_{k}\in\mathcal{L}^{1}(\mu)$ and $\int_{X}h_{k}\,d\mu=a\int_{X}f_{k}\,d\mu+b\int_{X}g_{k}\,d\mu$ (Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (vii)). By Step 2, $h\in\mathcal{L}^{1}(\mu;\mathbb{R}^{n})$ and, for every $k\in[n]$,

$$\Bigl(\int_{X}h\,d\mu\Bigr)_{k}=a\Bigl(\int_{X}f\,d\mu\Bigr)_{k}+b\Bigl(\int_{X}g\,d\mu\Bigr)_{k}=\Bigl(a\int_{X}f\,d\mu+b\int_{X}g\,d\mu\Bigr)_{k},$$

so $\int_{X}h\,d\mu=a\int_{X}f\,d\mu+b\int_{X}g\,d\mu$ (Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (iv)).

Proof of (iv). The same, with entries in place of components and the entrywise operations of Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (iv).

Proof of (v). Let $h=Cf$. For $i\in[m]$ and $x\in X$, $h_{i}(x)=\sum_{j=1}^{n}C_{ij}f_{j}(x)$ (Matrix-Vector Product (i)). Step 1, with $c_{j}=C_{ij}$, gives $h_{i}\in\mathcal{L}^{1}(\mu)$ and

$$\int_{X}h_{i}\,d\mu=\sum_{j=1}^{n}C_{ij}\int_{X}f_{j}\,d\mu=\sum_{j=1}^{n}C_{ij}\Bigl(\int_{X}f\,d\mu\Bigr)_{j}=\Bigl(C\int_{X}f\,d\mu\Bigr)_{i}$$

(Integrable Vector- and Matrix-Valued Functions and Their Integrals (ii), Matrix-Vector Product (i)). By Step 2, $h\in\mathcal{L}^{1}(\mu;\mathbb{R}^{m})$ and $\int_{X}h\,d\mu=C\int_{X}f\,d\mu$.

Proof of (vi). Let $G$ be the function $x\mapsto CF(x)$ into $\mathbb{R}^{r\times q}$ (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), The Matrix Product, Matrix-Vector Product, Transpose, Trace, Identity Matrix and Standard Basis Vectors Exist and Are Unique (i)). For $i\in[r]$, $j\in[q]$ and $x\in X$, $G_{ij}(x)=\sum_{a=1}^{p}C_{ia}F_{aj}(x)$, so Step 1, with $c_{a}=C_{ia}$, and Step 2 give $G\in\mathcal{L}^{1}(\mu;\mathbb{R}^{r\times q})$ and

$$\Bigl(\int_{X}G\,d\mu\Bigr)_{ij}=\sum_{a=1}^{p}C_{ia}\Bigl(\int_{X}F\,d\mu\Bigr)_{aj}=\Bigl(C\int_{X}F\,d\mu\Bigr)_{ij}$$

(Product of Real Matrices (i), Integrable Vector- and Matrix-Valued Functions and Their Integrals (iv)). Now $CFD(x)=G(x)D$, and for $i\in[r]$, $k\in[s]$, $(CFD)_{ik}(x)=\sum_{b=1}^{q}D_{bk}G_{ib}(x)$; Step 1, with $c_{b}=D_{bk}$, and Step 2 give $CFD\in\mathcal{L}^{1}(\mu;\mathbb{R}^{r\times s})$ and

$$\Bigl(\int_{X}CFD\,d\mu\Bigr)_{ik}=\sum_{b=1}^{q}\Bigl(\int_{X}G\,d\mu\Bigr)_{ib}D_{bk}=\Bigl(\Bigl(C\int_{X}F\,d\mu\Bigr)D\Bigr)_{ik}$$

(Product of Real Matrices (i), Integrable Vector- and Matrix-Valued Functions and Their Integrals (iv)).

Proof of (vii). For $x\in X$, $(u\cdot f)(x)=\sum_{k=1}^{n}u_{k}f_{k}(x)$ (The Dot Product on $\mathbb{R}^n$ (i)). Step 1, with $c_{k}=u_{k}$, gives $u\cdot f\in\mathcal{L}^{1}(\mu)$ and

$$\int_{X}u\cdot f\,d\mu=\sum_{k=1}^{n}u_{k}\int_{X}f_{k}\,d\mu=\sum_{k=1}^{n}u_{k}\Bigl(\int_{X}f\,d\mu\Bigr)_{k}=u\cdot\int_{X}f\,d\mu$$

(Integrable Vector- and Matrix-Valued Functions and Their Integrals (ii), The Dot Product on $\mathbb{R}^n$ (i)).

Step 3: scalar multiples. For $g\in\mathcal{L}^{1}(\mu)$ and $t\in\mathbb{R}$, the scalar multiple $tg$ has $tg(x)=t\,g(x)+0\,g(x)$, so $tg\in\mathcal{L}^{1}(\mu)$ and $\int_{X}tg\,d\mu=t\int_{X}g\,d\mu$ (Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (vii)).

Proof of (viii). As $f$ is measurable (Integrable Vector- and Matrix-Valued Functions and Their Integrals (i)), $\lVert f\rVert\in\mathcal{L}^{1}(\mu)$ by (i). Let $v=\int_{X}f\,d\mu$. By (vii) and Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (i),

$$\int_{X}v\cdot f\,d\mu=v\cdot v=\lVert v\rVert^{2}.$$

For $x\in X$, $v\cdot f(x)\le|v\cdot f(x)|\le\lVert v\rVert\,\lVert f(x)\rVert$ and $0\le\lVert f(x)\rVert$ (The Cauchy-Schwarz Inequality on $\mathbb{R}^n$ (i), Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (i)). With Step 3, monotonicity (Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (ix)) gives

$$\lVert v\rVert^{2}\le\lVert v\rVert\int_{X}\lVert f\rVert\,d\mu\qquad\text{and}\qquad0\le\int_{X}\lVert f\rVert\,d\mu .$$

As $\lVert v\rVert\ge0$, either $\lVert v\rVert=0$ and the second inequality concludes, or $\lVert v\rVert>0$ and dividing the first by $\lVert v\rVert$ gives $\lVert v\rVert\le\int_{X}\lVert f\rVert\,d\mu$ (Rules of Arithmetic and Order in an Ordered Field (vi), with $z=\lVert v\rVert$).

Proof of (ix). As $F$ is measurable (Integrable Vector- and Matrix-Valued Functions and Their Integrals (iii)), $\lVert F\rVert\in\mathcal{L}^{1}(\mu)$ by (ii). Let $A=\int_{X}F\,d\mu$, let $\xi\in\mathbb{R}^{q}$, and let $h_{\xi}$ be the function $x\mapsto F(x)\xi$ into $\mathbb{R}^{p}$ (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), The Matrix Product, Matrix-Vector Product, Transpose, Trace, Identity Matrix and Standard Basis Vectors Exist and Are Unique (ii)). For $i\in[p]$, $(h_{\xi})_{i}(x)=\sum_{j=1}^{q}\xi_{j}F_{ij}(x)$, so Step 1, with $c_{j}=\xi_{j}$, and Step 2 give $h_{\xi}\in\mathcal{L}^{1}(\mu;\mathbb{R}^{p})$ and $\int_{X}h_{\xi}\,d\mu=A\xi$ (Matrix-Vector Product (i), Integrable Vector- and Matrix-Valued Functions and Their Integrals (iv)). By (viii), $\lVert h_{\xi}\rVert\in\mathcal{L}^{1}(\mu)$ and $\lVert A\xi\rVert\le\int_{X}\lVert h_{\xi}\rVert\,d\mu$. For $x\in X$, $\lVert F(x)\xi\rVert\le\lVert\xi\rVert\,\lVert F(x)\rVert$ (Rules for the Operator Norm: Bound, Least Constant, Definiteness, Homogeneity, Triangle Inequality and Products (ii)), so Step 3 and monotonicity (Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (ix)) give

$$\lVert A\xi\rVert\le\Bigl(\int_{X}\lVert F\rVert\,d\mu\Bigr)\lVert\xi\rVert\qquad(\xi\in\mathbb{R}^{q}),$$

hence $\lVert A\rVert\le\int_{X}\lVert F\rVert\,d\mu$ (Rules for the Operator Norm: Bound, Least Constant, Definiteness, Homogeneity, Triangle Inequality and Products (iii)).