The inertia tensor of a finite bounded Borel measure about any point of ${\mathbb{R}}^3$

Let $\mathcal{B}({\mathbb{R}}^3)$ be the Borel sigma-algebra of ${\mathbb{R}}^3$ (The Borel sigma-algebra of a topological space), let $\nu$ be a measure on the measurable space $({\mathbb{R}}^3,\mathcal{B}({\mathbb{R}}^3))$ (Measures on sigma-algebras, Measure spaces) that is finite and bounded: $\nu ({\mathbb{R}}^3)<+\mathrm{\infty }$ and $\nu ({\mathbb{R}}^3\setminus {\bar{B}}_R(0))=0$ for some real $R>0$ (Open ball, closed ball and sphere in a metric space). Put $m:=\nu ({\mathbb{R}}^3)\in [0,+\mathrm{\infty })$; nonnegativity follows from the definition of measure Measures on sigma-algebras. If $m>0$, then $\nu$ is a mass distribution in the sense of Mass distributions on ${\mathbb{R}}^3$: total mass, first moment and inertia matrix. If $m=0$, then for every Borel set $E\subseteq {\mathbb{R}}^3$ one has $0\le \nu (E)\le \nu ({\mathbb{R}}^3)=0$, so $\nu$ is the zero measure. Integrals against $\nu$ are Lebesgue integrals as in Integral over a measurable subset and Integrable real and complex functions, and their integrals, defined componentwise for vector- and matrix-valued integrands. When $m>0$, recall that the inertia matrix of $\nu$ is

$I(\nu )=\int (\parallel \rho {\parallel }_2^2{I}_3-\rho {\rho }^{\mathsf{T}})d\nu (\rho )\in M_3(\mathbb{R}),$

with $I_3$ the identity matrix and entrywise integrals $I(\nu {)}_{jk}=\int (\parallel \rho {\parallel }_2^2{\delta }_{jk}-{\rho }_j{\rho }_k)d\nu$, where $\parallel \cdot {\parallel }_2$ and the matrix conventions are those of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$, The vector space $M_{m\times n}(F):=F^{m\times n}$ of $m$ by $n$ matrices over a field, with entrywise operations and Rectangular matrix multiplication and the identity matrix $I_n$, including zero-sized shapes.

Shifted integrand and inertia tensor about a point. For $a\in {\mathbb{R}}^3$ define the component functions

$G_a(\rho {)}_{jk}:=\parallel \rho -a{\parallel }_2^2{\delta }_{jk}-({\rho }_j-a_j)({\rho }_k-a_k)(j,k\in \{1,2,3\}),$

equivalently the matrix-valued function $G_a(\rho )=\parallel \rho -a{\parallel }_2^2{I}_3-(\rho -a)(\rho -a{)}^{\mathsf{T}}$, and define the inertia tensor of $\nu$ about $a$ $I_{\nu }(a)\in M_3(\mathbb{R})$ by its entries

$I_{\nu }(a{)}_{jk}:=\int G_a(\rho {)}_{jk}d\nu (\rho )\mathrm{.}$

Under the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) each entry is a finite real number; this is the content of Finiteness and a uniform bound for the entries of the inertia tensor about an arbitrary point ↗, which depends on this definition only for its notation. For $a=0$ the shifted integrand is the parent integrand $\parallel \rho {\parallel }_2^2{I}_3-\rho {\rho }^{\mathsf{T}}$. Thus $I_{\nu }(0)=I(\nu )$ when $m>0$; if $m=0$, the parent definition does not define $I(\nu )$, while the shifted integral remains defined and equals the zero matrix $0_{3\times 3}$ for every $a$.

Scope. The definition names a mathematical function $a\mapsto I_{\nu }(a)$ from ${\mathbb{R}}^3$ to $M_3(\mathbb{R})$ determined by any finite bounded Borel measure $\nu$ and its coordinate origin. For the zero measure this function is identically $0_{3\times 3}$. No physical body, unit system or measured mass is asserted. No translation identity for nonzero $a$ is asserted here; the centred, positive-total-mass case is proved separately on this page.

• The measure need not be atomless. Points, curves, surfaces and volumes are all admissible: $\nu$ is an arbitrary bounded finite Borel measure, so finite sums of Dirac measures, lines and surface measures embedded in ${\mathbb{R}}^3$ and absolutely continuous volume measures are all special cases, exactly as in Mass distributions on ${\mathbb{R}}^3$: total mass, first moment and inertia matrix.

• Translation of the point, not of the measure. The point $a$ only shifts the reference point at which the second moments are computed; the measure and its coordinate representation are unchanged, and $\mu (\nu )$ is not assumed to vanish unless a consumer states it.