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

In the settings of Measure Spaces, Borel Sets and Lebesgue Measure on the Real Line: Standing Notation and Euclidean Space and Real Matrices: Standing Notation, let $(X,\mathcal{F})$ be a measurable space and let $n,p,q\in\mathbb{N}$. Measurable maps from $X$ to $\mathbb{R}$, to $\mathbb{R}^{n}$ and to $\mathcal{M}_{p\times q}(\mathbb{R})$, for these or any other dimensions, are as in 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), and $\mathcal{L}=\mathrm{Meas}(X,\mathcal{F};\mathbb{R},\mathcal{B}(\mathbb{R}))$ is the set of measurable maps $X\to\mathbb{R}$, as in Measurable Map, and the Set of Measurable Maps (ii). For a map $g:X\to\mathbb{R}^{n}$ and $k\in[n]$, $g_{k}$ is its $k$-th component map, and for a map $G:X\to\mathcal{M}_{p\times q}(\mathbb{R})$, $i\in[p]$ and $j\in[q]$, $G_{ij}$ is its entry map.

(i) A map $g:X\to\mathbb{R}^{n}$ is measurable if and only if $g_{k}$ is measurable for every $k\in[n]$.

(ii) A map $G:X\to\mathcal{M}_{p\times q}(\mathbb{R})$ is measurable if and only if $G_{ij}$ is measurable for all $i\in[p]$ and $j\in[q]$.

(iii) Let $k\mapsto f_{k}$ be a map from $[n]$ to $\mathcal{L}$, and let $F:X\to\mathbb{R}^{n}$ be the map with $F(x)_{k}=f_{k}(x)$ for all $x\in X$ and $k\in[n]$, which exists and is unique by Component Maps of a Map into $\mathbb{R}^{q}$, Entry Maps of a Map into Matrices, and Maps Assembled from Them (iii); write $F(x)=(f_{1}(x),\dots,f_{n}(x))$. Let $l,s\in\mathbb{N}$, and let $\varphi$ be a map from $\mathbb{R}^{n}$ to $\mathbb{R}$, to $\mathbb{R}^{l}$ or to $\mathcal{M}_{l\times s}(\mathbb{R})$ that is continuous for the metrics $d_{E}$ on $\mathbb{R}^{n}$ and $d_{\mathbb{R}}$, $d_{E}$ or $d_{l\times s}$ on the target. Then the map $x\mapsto\varphi((f_{1}(x),\dots,f_{n}(x)))$ on $X$ is measurable.