Material regions, densities and their mass

A material region is a nonempty, bounded, open subset $B\subseteq {\mathbb{R}}^3$. This region predicate is choice-free and does not assume a density or measure.

For the following Lebesgue density and mass construction, assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (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 supplies the measure ${\lambda }_3$ on its Lebesgue sigma-algebra (Lebesgue measurable sets, the family $\mathcal{L}({\mathbb{R}}^n)$, and the restricted set function ${\lambda }_n$, Measures on sigma-algebras). Since $B$ is open, it is Lebesgue measurable by the Borel-measurability theorem (Assuming countable choice, every Borel subset of ${\mathbb{R}}^n$ is Lebesgue measurable). Put ${\mathcal{L}}_B:=\{E\cap B:E\in \mathcal{L}({\mathbb{R}}^3)\}$ and ${\lambda }_B(C):={\lambda }_3(C)$ for $C\in {\mathcal{L}}_B$. The trace family ${\mathcal{L}}_B$ is a sigma-algebra on $B$ (Sigma-algebras); countable additivity of ${\lambda }_B$ follows by applying that of ${\lambda }_3$ to disjoint trace-measurable sets, so $(B,{\mathcal{L}}_B,{\lambda }_B)$ is a measure space (Measures on sigma-algebras). In the displays below, ${\int }_B\rho d{\lambda }_3$ means the Lebesgue integral of $\rho$ on this measure space (Integrable real and complex functions, and their integrals). A density on $B$ is a measurable function

$\rho :B⟶[0,\mathrm{\infty })$

measurable as a map from $(B,{\mathcal{L}}_B)$ to $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ (Measurable spaces and measurable sets, A measurable function between measurable spaces, The Borel sigma-algebra of a topological space) and integrable on $(B,{\mathcal{L}}_B,{\lambda }_B)$; the mass of the pair $(B,\rho )$ is

$\mathrm{mass}(B,\rho ):={\int }_B\rho d{\lambda }_3\in [0,\mathrm{\infty })\mathrm{.}$

For this scalar function, bounded means that some $M<\mathrm{\infty }$ satisfies $0\le \rho (X)\le M$ for every $X\in B$. Continuous means that for every $X\in B$ and every $\epsilon >0$ there is $\delta >0$ such that $Y\in B$ and $\mathrm{\parallel }X-Y{\mathrm{\parallel }}_2<\delta$ imply $\mathrm{\mid }\rho (X)-\rho (Y)\mathrm{\mid }<\epsilon$. For $k\ge 1$, of class $C^k$ on $B$ means that every iterated coordinate partial derivative of order at most $k$ exists on $B$ and is continuous there; no extension to $\mathrm{\partial }B$ is required unless a consuming statement says so. A measured material region is a triple $(B,\rho ,m)$ with $m=\mathrm{mass}(B,\rho )$ recorded.

• Only measurability and integrability on the restricted Lebesgue measure space $(B,{\mathcal{L}}_B,{\lambda }_B)$, together with the displayed nonnegative codomain, are part of the density definition. A density is not required to be positive, bounded, continuous, or supported away from $\mathrm{\partial }B$, and the definition asserts no conservation law: the mass is a number attached to $(B,\rho )$.

• Boundary regularity is a declared hypothesis, not part of the word "region": when a statement needs it, it says that the boundary $\mathrm{\partial }B$ is a $C^1$ embedded surface (or a smooth/rectifiable hypersurface) and states the surface-measure conventions used. The local transport and balance items of this collection state their own boundary hypotheses; none is inherited from this definition.

• The unit of $\rho$ is mass per unit volume only after a unit system is fixed. The mathematical object here is a function and a number; no physical identification of the number $m$ with a measured mass is made, and no empirical claim is attached.

• The statement "$\rho$ is mass per volume" is the interpretation convention $m={\int }_B\rho d{\lambda }_3$ recorded above; it is a definition of the interpretation, not a physical law. In particular, if $\rho$ is a summable function with a discontinuity across a surface, the integral above is still the Lebesgue integral and no interfacial mass is implied unless it is separately declared.