Finite measurable squared-error quantizer and nearest-center rule

Let $d,k\ge 1$, let $X\subseteq {\mathbb{R}}^d$ be Borel, and give $X$ the relative Borel sigma algebra $\mathcal{B}(X)=\{X\cap B:B\text{ is in the sigma algebra generated by Euclidean open sets in }{\mathbb{R}}^d\}$. Let $\mu$ be a probability measure on $\mathcal{B}(X)$ with ${\int }_X\mathrm{\parallel }x{\mathrm{\parallel }}_2^{2}d\mu (x)<\mathrm{\infty }$, and let $c_1,\dots ,c_k\in {\mathbb{R}}^d$ be reproduction points, where $\mathrm{\parallel }x{\mathrm{\parallel }}_2^2:={\sum }_{\mathrm{\ell }=1}^d{x}_{\mathrm{\ell }}^2$. A finite squared-error quantizer is a measurable map $Q:X\to \{1,\dots ,k\}$, with the finite index set given its discrete sigma algebra; its reproduction at $x$ is $c_{Q(x)}$ and its distortion is

$D_{\mu }(Q;c_1,\dots ,c_k):={\int }_X\mathrm{\parallel }x-c_{Q(x)}{\mathrm{\parallel }}_2^{2}d\mu (x)\mathrm{.}$

This is the expected single-letter distortion of Single-letter and block distortion for the measurable distortion $d(x,c)=\mathrm{\parallel }x-c{\mathrm{\parallel }}_2^2$. For a fixed ordered center tuple $c=(c_1,\dots ,c_k)$, the nearest-center rule assigns $x$ to the least index minimizing $\mathrm{\parallel }x-c_j{\mathrm{\parallel }}_2^2$. Its cell at index $i$ is

$C_i(c):=X\cap {\bigcap }_{j<i}\{x:\mathrm{\parallel }x-c_i{\mathrm{\parallel }}_2^2<\mathrm{\parallel }x-c_j{\mathrm{\parallel }}_2^2\}\cap {\bigcap }_{j>i}\{x:\mathrm{\parallel }x-c_i{\mathrm{\parallel }}_2^2\le \mathrm{\parallel }x-c_j{\mathrm{\parallel }}_2^2\}\mathrm{.}$

The convention assigns every tie to the smallest index. A general finite measurable quantizer need not use this nearest-center partition; in particular, a compander partition below is specified by its own interval boundaries. When $X={\mathbb{R}}^d$, this is the ordinary full-domain nearest-center quantizer.