Intersection-volume conservative remapping

Assume Countable Choice. Let $\mathrm{\Omega }\subset {\mathbb{R}}^d$ be a bounded Lebesgue-measurable domain of finite positive measure. Let $\mathcal{S}=(S_i{)}_{i=1}^m$ and $\mathcal{T}=(T_j{)}_{j=1}^n$ be finite measurable partitions of $\mathrm{\Omega }$ up to null sets, with disjoint interiors modulo null sets and positive cell volumes. Define exact overlap weights $w_{ji}=\mathrm{\mid }{T}_j\cap S_i\mathrm{\mid }$. For source constants $v_i$, the intersection-volume remap is the target piecewise-constant field $Rv$ whose value on $T_j$ is $(Rv{)}_j=\mathrm{\mid }{T}_j{\mathrm{\mid }}^{-1}{\sum }_i{w}_{ji}{v}_i$. All weights are exact measures; an approximate clipping weight is not silently substituted.

The measurable sets and restricted set function are defined in Lebesgue measurable sets, the family $\mathcal{L}({\mathbb{R}}^n)$, and the restricted set function ${\lambda }_n$. Under Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), Assuming countable choice, $\mathcal{L}({\mathbb{R}}^n)$ is a sigma-algebra containing every elementary set and ${\lambda }_n$ is a complete measure extending elementary volume makes that family a sigma-algebra and the set function a measure. Thus every $T_j\cap S_i$ is measurable, and its measure is finite because it lies in $\mathrm{\Omega }$; every target denominator is finite and positive. Stable cell identifiers fix the finite summation order but do not affect the real-valued result. A target cell with zero volume is excluded because its average is undefined. The remap is a finite-volume $L^2$ projection once the conservation and orthogonality theorem below is proved.