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

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 $n,p,q,r\in\mathbb{N}$. 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). The sum, the scalar multiple, the dot product and the Euclidean norm on $\mathbb{R}^{n}$, and the sum, the scalar multiple, the matrix product, the matrix-vector product, the transpose and the operator norm of real matrices, are as in Euclidean Space and Real Matrices: Standing Notation (ii), Euclidean Space and Real Matrices: Standing Notation (iii) and Euclidean Space and Real Matrices: Standing Notation (vi).

(i) Let $g,h:X\to\mathbb{R}^{n}$ be measurable and let $c\in\mathbb{R}$. Then the maps $x\mapsto g(x)+h(x)$ and $x\mapsto c\,g(x)$ from $X$ to $\mathbb{R}^{n}$, and the map $x\mapsto g(x)\cdot h(x)$ from $X$ to $\mathbb{R}$, are measurable.

(ii) Let $G,H:X\to\mathcal{M}_{p\times q}(\mathbb{R})$, $K:X\to\mathcal{M}_{q\times r}(\mathbb{R})$ and $v:X\to\mathbb{R}^{q}$ be measurable, and let $c\in\mathbb{R}$. Then the maps $x\mapsto G(x)+H(x)$ and $x\mapsto c\,G(x)$ from $X$ to $\mathcal{M}_{p\times q}(\mathbb{R})$, $x\mapsto G(x)K(x)$ from $X$ to $\mathcal{M}_{p\times r}(\mathbb{R})$, $x\mapsto G(x)^{\top}$ from $X$ to $\mathcal{M}_{q\times p}(\mathbb{R})$, and $x\mapsto G(x)v(x)$ from $X$ to $\mathbb{R}^{p}$ are measurable.

(iii) Let $g:X\to\mathbb{R}^{n}$ and $G:X\to\mathcal{M}_{p\times q}(\mathbb{R})$ be measurable. Then the maps $x\mapsto\lVert g(x)\rVert$, with the Euclidean norm, and $x\mapsto\lVert G(x)\rVert$, with the operator norm, from $X$ to $\mathbb{R}$ are measurable.