The Set of Coordinate Preimages of Borel Sets in ℝⁿ and of Entry Preimages in the Real Matrices

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 $n,p,q\in\mathbb{N}$, let $\mathcal{B}(\mathbb{R})$ be the Borel $\sigma$-algebra of Borel Sigma-Algebra of a Metric Space (ii), and let $\mathcal{P}(\mathbb{R}^{n})$ and $\mathcal{P}(\mathcal{M}_{p\times q}(\mathbb{R}))$ be the power sets of $\mathbb{R}^{n}$ and of $\mathcal{M}_{p\times q}(\mathbb{R})$. For $i\in[n]$, $\pi_{i}:\mathbb{R}^{n}\to\mathbb{R}$ is the $i$-th component map of the identity $\mathrm{id}_{\mathbb{R}^{n}}$, so $\pi_{i}(z)=z_{i}$; for $i\in[p]$ and $j\in[q]$, $E_{ij}:\mathcal{M}_{p\times q}(\mathbb{R})\to\mathbb{R}$ is the entry map of the identity $\mathrm{id}_{\mathcal{M}_{p\times q}(\mathbb{R})}$ in row $i$ and column $j$, so $E_{ij}(A)=A_{ij}$. Preimages are as in Sets and Maps: Ordinary Notation (vii).

(i) There is exactly one set $\mathcal{C}$ with $\mathcal{C}\subseteq\mathcal{P}(\mathbb{R}^{n})$ such that, for every set $C$: $C\in\mathcal{C}$ if and only if $C=\pi_{i}^{-1}(B)$ for some $i\in[n]$ and some $B\in\mathcal{B}(\mathbb{R})$. It is written $\{\pi_{i}^{-1}(B):i\in[n],\ B\in\mathcal{B}(\mathbb{R})\}$.

(ii) There is exactly one set $\mathcal{C}'$ with $\mathcal{C}'\subseteq\mathcal{P}(\mathcal{M}_{p\times q}(\mathbb{R}))$ such that, for every set $C$: $C\in\mathcal{C}'$ if and only if $C=E_{ij}^{-1}(B)$ for some $i\in[p]$, some $j\in[q]$ and some $B\in\mathcal{B}(\mathbb{R})$. It is written $\{E_{ij}^{-1}(B):i\in[p],\ j\in[q],\ B\in\mathcal{B}(\mathbb{R})\}$.