Linear statistics on Euclidean data are Borel-measurable

Given: An integer $n\ge 1$, a vector $a\in {\mathbb{R}}^n$, and the standard Euclidean metrics and their Borel sigma-algebras on ${\mathbb{R}}^n$ and $\mathbb{R}$.

Every linear map $L:{\mathbb{R}}^m\to {\mathbb{R}}^k$, with $m,k\ge 1$, is bounded: for some $K\ge 0$, $\mathrm{\parallel }Lh{\mathrm{\parallel }}_2\le K\mathrm{\parallel }h{\mathrm{\parallel }}_2$ for every $h$. Linearity means preservation of linear combinations by A linear map $L:{\mathbb{R}}^m\to {\mathbb{R}}^n$ in Euclidean coordinates, and the bound is Every Euclidean linear map has a unique matrix and satisfies $\mathrm{\parallel }Lh{\mathrm{\parallel }}_2\le K\mathrm{\parallel }h{\mathrm{\parallel }}_2$ for some $K\ge 0$.

On ${\mathbb{R}}^m$ for $m\ge 1$, the norm from the Euclidean inner product agrees with $d_2$. The inner product is bilinear, and ${\mathbb{R}}^m$ consists of functions on $\{0,\dots ,m-1\}$ with pointwise vector operations. The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$, ${\mathbb{R}}^n$ as the set of functions $n\to \mathbb{R}$, and $d_1$, $d_2$, $d_{\mathrm{\infty }}$ are metrics on it.

For every $m\ge 1$, $d_2$ is a metric on ${\mathbb{R}}^m$ by ${\mathbb{R}}^n$ as the set of functions $n\to \mathbb{R}$, and $d_1$, $d_2$, $d_{\mathrm{\infty }}$ are metrics on it. For any metric space, its metric-open sets form a topology: the empty set and the whole space are open by The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement, and arbitrary unions and finite intersections are open by Arbitrary unions and finite intersections of open sets are open, open balls are open and closed balls are closed. The balls $B(x,r)$ used below are as defined by Open ball, closed ball and sphere in a metric space. Thus the metric topology gives a topological space by Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison, and its Borel sigma-algebra is generated by those open sets by The Borel sigma-algebra of a topological space.

A map of topological spaces is continuous when, for each point and each open target neighborhood of its image, it maps some open source neighborhood of the point into that target neighborhood. Continuity of a map of topological spaces at a point and globally.

A continuous map between topological spaces has Borel preimages of Borel sets. A continuous map has Borel preimages of Borel sets.

A map between measurable spaces is measurable when the preimage of every measurable target set is measurable in the source. A measurable function between measurable spaces.