Proof of Sums, Products, Transposes, Dot Products and Norms of Measurable Vector- and Matrix-Valued Maps

Let $\mathcal{L}=\mathrm{Meas}(X,\mathcal{F};\mathbb{R},\mathcal{B}(\mathbb{R}))$ be the set of measurable maps $X\to\mathbb{R}$, whose elements are exactly these maps by Measurable Map, and the Set of Measurable Maps (ii). For a map $u$ from $X$ to some $\mathbb{R}^{k}$ and $l\in[k]$, $u_{l}$ is its $l$-th component map, and for a map $A$ from $X$ to some $\mathcal{M}_{k\times k'}(\mathbb{R})$, $i\in[k]$ and $j\in[k']$, $A_{ij}$ is its entry map. Throughout, a map into Euclidean space is measurable if and only if all its component maps are, by Vector- and Matrix-Valued Measurable Maps: Components, Entries and Continuous Images (i), and a map into the real matrices is measurable if and only if all its entry maps are, by Vector- and Matrix-Valued Measurable Maps: Components, Entries and Continuous Images (ii); both are used for whichever dimensions occur.

Proof of (i). Let $g$, $h$ and $c$ be as there. For $k\in[n]$, the $k$-th component maps of $x\mapsto g(x)+h(x)$ and of $x\mapsto c\,g(x)$ are $g_{k}+h_{k}$ and $c\,g_{k}$, by Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (iv); $g_{k}$ and $h_{k}$ are measurable by Vector- and Matrix-Valued Measurable Maps: Components, Entries and Continuous Images (i), so these are measurable by Arithmetic of Measurable Real- and [0,∞]-Valued Maps (i), and the two maps are measurable by Vector- and Matrix-Valued Measurable Maps: Components, Entries and Continuous Images (i). By The Dot Product on $\mathbb{R}^n$ (i), $g(x)\cdot h(x)=\sum_{k=1}^{n}g_{k}(x)h_{k}(x)$ for $x\in X$, the sum being the same whether read as the iterated sum of Arithmetic of Measurable Real- and [0,∞]-Valued Maps (ii) or as the sum over $[n]$ (Natural Numbers as Indices: ℕ ⊆ ℕ₀, 1 ∈ ℕ₀, and the Two Readings of an Operation from 1 to n (v)), and $k\mapsto g_{k}h_{k}$ is a map from $[n]$ to $\mathcal{L}$, as follows. Let $\pi_{1}:[n]\times X\to[n]$ and $\pi_{2}:[n]\times X\to X$ be the projections of Projection, Evaluation and Currying Maps (i), and $\mathrm{ev}:\mathbb{R}^{[n]}\times[n]\to\mathbb{R}$ the evaluation map of Projection, Evaluation and Currying Maps (ii), where $\mathbb{R}^{[n]}=\mathbb{R}^{n}$ by Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (i). By Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (iii), applied to $g\circ\pi_{2}$, $\pi_{1}$ and $\mathrm{ev}$, there is a map $A:[n]\times X\to\mathbb{R}$ with $A(k,x)=g(x)(k)=g(x)_{k}=g_{k}(x)$, by Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (ii) and Component Maps and Entry Maps (i); likewise a map $B:[n]\times X\to\mathbb{R}$ with $B(k,x)=h_{k}(x)$; and, applied to $A$, $B$ and the multiplication $\mathbb{R}\times\mathbb{R}\to\mathbb{R}$, a map $\Gamma:[n]\times X\to\mathbb{R}$ with $\Gamma(k,x)=g_{k}(x)h_{k}(x)$. By Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (iv), there is a map $\Phi:[n]\to\mathbb{R}^{X}$ with $\Phi(k)(x)=\Gamma(k,x)$, so that $\Phi(k)$ is the map $g_{k}h_{k}$ for every $k\in[n]$. Each $g_{k}h_{k}$ is measurable by Vector- and Matrix-Valued Measurable Maps: Components, Entries and Continuous Images (i) and Arithmetic of Measurable Real- and [0,∞]-Valued Maps (i), so $\Phi$ is a map from $[n]$ to $\mathcal{L}$. Hence $x\mapsto g(x)\cdot h(x)$ is measurable by Arithmetic of Measurable Real- and [0,∞]-Valued Maps (ii).

Proof of (ii). Let $G$, $H$, $K$, $v$ and $c$ be as there. For $i\in[p]$ and $j\in[q]$, the entry maps of $x\mapsto G(x)+H(x)$ and of $x\mapsto c\,G(x)$ in row $i$ and column $j$ are $G_{ij}+H_{ij}$ and $c\,G_{ij}$, by Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (iv); $G_{ij}$ and $H_{ij}$ are measurable by Vector- and Matrix-Valued Measurable Maps: Components, Entries and Continuous Images (ii), so these are measurable by Arithmetic of Measurable Real- and [0,∞]-Valued Maps (i), and the two maps are measurable by Vector- and Matrix-Valued Measurable Maps: Components, Entries and Continuous Images (ii). For $i\in[p]$ and $j\in[r]$, the entry map of $x\mapsto G(x)K(x)$ in row $i$ and column $j$ is $x\mapsto\sum_{\ell=1}^{q}G_{i\ell}(x)K_{\ell j}(x)$, by Product of Real Matrices (i), the sum being the same in either reading (Natural Numbers as Indices: ℕ ⊆ ℕ₀, 1 ∈ ℕ₀, and the Two Readings of an Operation from 1 to n (v)); and $\ell\mapsto G_{i\ell}K_{\ell j}$ is a map from $[q]$ to $\mathcal{L}$, as in the proof of (i). Let $\pi_{1}:[q]\times X\to[q]$ and $\pi_{2}:[q]\times X\to X$ be the projections of Projection, Evaluation and Currying Maps (i), and let $\theta_{i}:[q]\times X\to[p]$ and $\theta'_{j}:[q]\times X\to[r]$ be the maps with the constant values $i$ and $j$ (Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (i)). By Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (ii), applied to $\theta_{i}$ and $\pi_{1}$ and to $\pi_{1}$ and $\theta'_{j}$, there are maps $\sigma:[q]\times X\to[p]\times[q]$ and $\sigma':[q]\times X\to[q]\times[r]$ with $\sigma(\ell,x)=(i,\ell)$ and $\sigma'(\ell,x)=(\ell,j)$. By Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (iii), applied to $G\circ\pi_{2}$, $\sigma$ and the evaluation map $\mathrm{ev}:\mathbb{R}^{[p]\times[q]}\times([p]\times[q])\to\mathbb{R}$ of Projection, Evaluation and Currying Maps (ii), there is a map $A:[q]\times X\to\mathbb{R}$ with $A(\ell,x)=G(x)(i,\ell)=G(x)_{i\ell}=G_{i\ell}(x)$, by Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (i), Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (ii) and Component Maps and Entry Maps (ii); likewise, from $K\circ\pi_{2}$, $\sigma'$ and the evaluation map of $\mathbb{R}^{[q]\times[r]}$, a map $B:[q]\times X\to\mathbb{R}$ with $B(\ell,x)=K_{\ell j}(x)$; and, applied to $A$, $B$ and the multiplication of $\mathbb{R}$, a map $\Gamma:[q]\times X\to\mathbb{R}$ with $\Gamma(\ell,x)=G_{i\ell}(x)K_{\ell j}(x)$. By Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (iv), there is a map $\Phi:[q]\to\mathbb{R}^{X}$ with $\Phi(\ell)(x)=\Gamma(\ell,x)$, so $\Phi(\ell)=G_{i\ell}K_{\ell j}$ for every $\ell\in[q]$. Each $G_{i\ell}K_{\ell j}$ is measurable by Vector- and Matrix-Valued Measurable Maps: Components, Entries and Continuous Images (ii) and Arithmetic of Measurable Real- and [0,∞]-Valued Maps (i), so $\Phi$ is a map from $[q]$ to $\mathcal{L}$, and this entry map is measurable by Arithmetic of Measurable Real- and [0,∞]-Valued Maps (ii); hence $x\mapsto G(x)K(x)$ is measurable by Vector- and Matrix-Valued Measurable Maps: Components, Entries and Continuous Images (ii). For $i\in[q]$ and $j\in[p]$, the entry map of $x\mapsto G(x)^{\top}$ in row $i$ and column $j$ is $G_{ji}$, by Transpose of a Real Matrix (i), which is measurable by Vector- and Matrix-Valued Measurable Maps: Components, Entries and Continuous Images (ii); so $x\mapsto G(x)^{\top}$ is measurable by Vector- and Matrix-Valued Measurable Maps: Components, Entries and Continuous Images (ii). For $i\in[p]$, the $i$-th component map of $x\mapsto G(x)v(x)$ is $x\mapsto\sum_{\ell=1}^{q}G_{i\ell}(x)v_{\ell}(x)$, by Matrix-Vector Product (i), the sum being the same in either reading (Natural Numbers as Indices: ℕ ⊆ ℕ₀, 1 ∈ ℕ₀, and the Two Readings of an Operation from 1 to n (v)); and $\ell\mapsto G_{i\ell}v_{\ell}$ is a map from $[q]$ to $\mathcal{L}$, in the same way: with $\pi_{1}$, $\pi_{2}$, $\theta_{i}$, $\sigma$ and $A$ built as above for this $i$, Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (iii), applied to $v\circ\pi_{2}$, $\pi_{1}$ and the evaluation map $\mathbb{R}^{[q]}\times[q]\to\mathbb{R}$, gives a map $B':[q]\times X\to\mathbb{R}$ with $B'(\ell,x)=v(x)_{\ell}=v_{\ell}(x)$, by Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (ii) and Component Maps and Entry Maps (i), and, applied to $A$, $B'$ and the multiplication of $\mathbb{R}$, a map $\Gamma':[q]\times X\to\mathbb{R}$ with $\Gamma'(\ell,x)=G_{i\ell}(x)v_{\ell}(x)$; Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (iv) gives a map $\Phi':[q]\to\mathbb{R}^{X}$ with $\Phi'(\ell)=G_{i\ell}v_{\ell}$ for every $\ell\in[q]$. Each $G_{i\ell}v_{\ell}$ is measurable by Vector- and Matrix-Valued Measurable Maps: Components, Entries and Continuous Images (ii), Vector- and Matrix-Valued Measurable Maps: Components, Entries and Continuous Images (i) and Arithmetic of Measurable Real- and [0,∞]-Valued Maps (i), so $\Phi'$ is a map from $[q]$ to $\mathcal{L}$, and this component map is measurable by Arithmetic of Measurable Real- and [0,∞]-Valued Maps (ii); hence $x\mapsto G(x)v(x)$ is measurable by Vector- and Matrix-Valued Measurable Maps: Components, Entries and Continuous Images (i).

Proof of (iii). Let $g$ and $G$ be as there. The identity maps of $(\mathbb{R}^{n},d_{E})$ and of $(\mathcal{M}_{p\times q}(\mathbb{R}),d_{p\times q})$ are continuous on the whole space by Continuity of Identity Maps, Compositions and Reciprocals (i). Hence, by Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (iii) and Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (v), applied to these identity maps with $S$ the whole space, the Euclidean norm $z\mapsto\lVert z\rVert$ on $\mathbb{R}^{n}$, a map $\mathbb{R}^{n}\to\mathbb{R}$ by The Summation Map on Real-Valued Maps over a Finite Set, and the Dot Product, Euclidean Norm and Euclidean Distance as Maps (iii), and the operator norm $A\mapsto\lVert A\rVert$ on $\mathcal{M}_{p\times q}(\mathbb{R})$ are continuous into $(\mathbb{R},d_{\mathbb{R}})$.

By Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (iv), with the Borel $\sigma$-algebras of Borel Sigma-Algebra of a Metric Space (iii), Borel Sigma-Algebra of a Metric Space (iv) and Borel Sigma-Algebra of a Metric Space (ii), the Euclidean norm on $\mathbb{R}^{n}$ is $\mathcal{B}(\mathbb{R}^{n})/\mathcal{B}(\mathbb{R})$-measurable and the operator norm on $\mathcal{M}_{p\times q}(\mathbb{R})$ is $\mathcal{B}(\mathbb{R}^{p\times q})/\mathcal{B}(\mathbb{R})$-measurable. As $g$ is $\mathcal{F}/\mathcal{B}(\mathbb{R}^{n})$-measurable by Measurable Map, and the Set of Measurable Maps (iv) and $G$ is $\mathcal{F}/\mathcal{B}(\mathbb{R}^{p\times q})$-measurable by Measurable Map, and the Set of Measurable Maps (v), the composites of the two norms with $g$ and with $G$, which are the maps $x\mapsto\lVert g(x)\rVert$ and $x\mapsto\lVert G(x)\rVert$ (Basic Properties of Functions: Equality, Composition, Identity, Inverse and Restriction (ii)), are $\mathcal{F}/\mathcal{B}(\mathbb{R})$-measurable by Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (ii), that is, measurable by Measurable Map, and the Set of Measurable Maps (iii).