Proof of Vector- and Matrix-Valued Measurable Maps: Components, Entries and Continuous Images

By Measurable Map, and the Set of Measurable Maps (iii), Measurable Map, and the Set of Measurable Maps (iv) and Measurable Map, and the Set of Measurable Maps (v), a map from $X$ to $\mathbb{R}$, to $\mathbb{R}^{n}$ or to $\mathcal{M}_{p\times q}(\mathbb{R})$ is measurable when it is measurable with respect to $\mathcal{F}$ and $\mathcal{B}(\mathbb{R})$, $\mathcal{B}(\mathbb{R}^{n})$ or $\mathcal{B}(\mathbb{R}^{p\times q})$ respectively.

Components and entries. For $k\in[n]$ let $\pi_{k}:\mathbb{R}^{n}\to\mathbb{R}$ be the $k$-th component map of the identity map of $\mathbb{R}^{n}$ (Component Maps and Entry Maps (i)), so $\pi_{k}(z)=z_{k}$, and let $\mathcal{C}=\{\pi_{k}^{-1}(B):k\in[n],\ B\in\mathcal{B}(\mathbb{R})\}$; by The Set of Coordinate Preimages of Borel Sets in ℝⁿ and of Entry Preimages in the Real Matrices (i), $\mathcal{C}$ is a set, a subset of $\mathcal{P}(\mathbb{R}^{n})$, whose elements are exactly the sets $\pi_{k}^{-1}(B)$ with $k\in[n]$ and $B\in\mathcal{B}(\mathbb{R})$. By The Borel Sets of ℝⁿ and of the Real Matrices Are Generated by the Coordinates (iii), $\mathcal{B}(\mathbb{R}^{n})=\sigma(\mathcal{C})$. Let $g:X\to\mathbb{R}^{n}$. For $k\in[n]$ and $B\in\mathcal{B}(\mathbb{R})$ we have $g_{k}(x)=g(x)_{k}=\pi_{k}(g(x))$ for $x\in X$, hence $g_{k}^{-1}(B)=g^{-1}(\pi_{k}^{-1}(B))$. If $g$ is measurable, then $\pi_{k}^{-1}(B)\in\mathcal{C}\subseteq\sigma(\mathcal{C})=\mathcal{B}(\mathbb{R}^{n})$ by Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (iii), so $g_{k}^{-1}(B)\in\mathcal{F}$ by Measurable Map, and the Set of Measurable Maps (i); thus every $g_{k}$ is measurable. Conversely, if every $g_{k}$ is measurable, then for $C\in\mathcal{C}$, say $C=\pi_{k}^{-1}(B)$, we get $g^{-1}(C)=g_{k}^{-1}(B)\in\mathcal{F}$; so $g$ is measurable by Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (i), applied with the generating set $\mathcal{C}\subseteq\mathcal{P}(\mathbb{R}^{n})$ of $\mathcal{B}(\mathbb{R}^{n})$. This proves (i).

The same argument proves (ii): for $i\in[p]$ and $j\in[q]$ let $E_{ij}:\mathcal{M}_{p\times q}(\mathbb{R})\to\mathbb{R}$ be the entry map of the identity map of $\mathcal{M}_{p\times q}(\mathbb{R})$ (Component Maps and Entry Maps (ii)), so $E_{ij}(A)=A_{ij}$; let $\mathcal{C}'=\{E_{ij}^{-1}(B):i\in[p],\ j\in[q],\ B\in\mathcal{B}(\mathbb{R})\}$, by The Set of Coordinate Preimages of Borel Sets in ℝⁿ and of Entry Preimages in the Real Matrices (ii) a set, a subset of $\mathcal{P}(\mathcal{M}_{p\times q}(\mathbb{R}))$, whose elements are exactly the sets $E_{ij}^{-1}(B)$ with $i\in[p]$, $j\in[q]$ and $B\in\mathcal{B}(\mathbb{R})$; by The Borel Sets of ℝⁿ and of the Real Matrices Are Generated by the Coordinates (iv), $\mathcal{B}(\mathbb{R}^{p\times q})=\sigma(\mathcal{C}')$; and for $G:X\to\mathcal{M}_{p\times q}(\mathbb{R})$ we have $G_{ij}^{-1}(B)=G^{-1}(E_{ij}^{-1}(B))$, since $G_{ij}(x)=G(x)_{ij}=E_{ij}(G(x))$. The two directions then follow from Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (iii) and from Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (i), applied with the generating set $\mathcal{C}'$, exactly as above.

Continuous images. Let $k\mapsto f_{k}$, $F$, $l$, $s$ and $\varphi$ be as in (iii). By Component Maps of a Map into $\mathbb{R}^{q}$, Entry Maps of a Map into Matrices, and Maps Assembled from Them (iii), the component maps of $F$ are $F_{k}=f_{k}$, which are measurable; so $F$ is measurable by (i). The target of $\varphi$ is $(\mathbb{R},d_{\mathbb{R}})$, $(\mathbb{R}^{l},d_{E})$ or $(\mathcal{M}_{l\times s}(\mathbb{R}),d_{l\times s})$, whose Borel $\sigma$-algebra is $\mathcal{B}(\mathbb{R})$, $\mathcal{B}(\mathbb{R}^{l})$ or $\mathcal{B}(\mathbb{R}^{l\times s})$, and $\mathcal{B}(\mathbb{R}^{n})$ is the Borel $\sigma$-algebra of $(\mathbb{R}^{n},d_{E})$, by Borel Sigma-Algebra of a Metric Space (ii), Borel Sigma-Algebra of a Metric Space (iii) and Borel Sigma-Algebra of a Metric Space (iv). Hence $\varphi$ is measurable with respect to $\mathcal{B}(\mathbb{R}^{n})$ and the Borel $\sigma$-algebra of its target, by Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (iv), and $\varphi\circ F$ is measurable by Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (ii). Its value at $x$ is $\varphi((f_{1}(x),\dots,f_{n}(x)))$, which proves (iii).