The finite nearest-center quantizer is Borel with finite distortion

For fixed $i,j$, the difference $\mathrm{\parallel }x-c_i{\mathrm{\parallel }}_2^2-\mathrm{\parallel }x-c_j{\mathrm{\parallel }}_2^2$ is a continuous polynomial in the coordinates of $x$. Therefore its strict-negative set is open and its nonpositive set is closed; the finite intersections defining each $C_i(c)$ in [F1] are Borel.

The finite list of distances has a minimum at each $x\in X$, and the least minimizing index is unique. The tie inequalities in [F1] therefore make the cells a disjoint partition of $X$; the inverse image of each singleton index is relatively Borel by step 1.1, so the finite-valued index map is Borel.

For each coordinate, $(x_{\mathrm{\ell }}-c_{i,\mathrm{\ell }}{)}^2\le 2x_{\mathrm{\ell }}^2+2c_{i,\mathrm{\ell }}^2$, since $-2x_{\mathrm{\ell }}{c}_{i,\mathrm{\ell }}\le x_{\mathrm{\ell }}^2+c_{i,\mathrm{\ell }}^2$. Summing coordinates and integrating over the partition gives $D_{\mu }(Q_c;c)\le 2\int \mathrm{\parallel }x{\mathrm{\parallel }}_2^{2}d\mu (x)+2{\max }_{1\le i\le k}\mathrm{\parallel }{c}_i{\mathrm{\parallel }}_2^2<\mathrm{\infty }$, so the distortion is well-defined and finite.