Finiteness of the moments of a bounded finite Borel measure on ${\mathbb{R}}^3$

Each coordinate function $\rho \mapsto {\rho }_j$ is continuous, hence Borel measurable by [L1]; the products $\rho \mapsto {\rho }_j{\rho }_k$ are then measurable by the product clause of [L1].

Let $N:={\mathbb{R}}^3\setminus {\bar{B}}_R(0)$, a Borel set by [L6], with $\nu (N)=0$. We justify the almost-everywhere comparison from the definition of the nonnegative integral: if a nonnegative measurable $f$ satisfies $f\le B$ on $N^c$ for a finite constant $B\ge 0$, every nonnegative simple minorant $s\le f$ has the same simple integral as $s{\mathbf{1}}_{N^c}$. Indeed, in a finite disjoint level-set representation of $s$, the contribution of each intersection with $N$ is zero, while $\nu (A)=\nu (A\setminus N)$ for every measurable level set $A$. Pointwise $s{\mathbf{1}}_{N^c}\le B$, so [L2] and [L3] give $\int sd\nu \le B\nu ({\mathbb{R}}^3)$; taking the supremum over simple minorants gives $\int fd\nu \le B\nu ({\mathbb{R}}^3)$ by The nonnegative Lebesgue integral. Apply this with $f=\mathrm{\mid }{\rho }_j\mathrm{\mid }$, $B=R$, and with $f=\mathrm{\mid }{\rho }_j{\rho }_k\mathrm{\mid }$, $B=R^2$: measurability follows from step 1.1 and [L1], and [L5] gives the bounds on $N^c$. Thus $\int \mathrm{\mid }{\rho }_j\mathrm{\mid }d\nu \le R\nu ({\mathbb{R}}^3)$ and $\int \mathrm{\mid }{\rho }_j{\rho }_k\mathrm{\mid }d\nu \le R^2\nu ({\mathbb{R}}^3)$.

The bounds of step 2.1 are finite because $\nu$ is finite, so each of the functions ${\rho }_j$ and ${\rho }_j{\rho }_k$ has finite integral of its modulus, hence is $\nu$-integrable by [L4].

Each coordinate integrand is integrable by step 3.1. Each inertia integrand $g_{jk}(\rho ):=\mathrm{\parallel }\rho {\mathrm{\parallel }}_2^2{\delta }_{jk}-{\rho }_j{\rho }_k$ is measurable by [L1], and on $N^c$ its modulus is at most $2R^2$ by [L5]. The simple-minorant comparison proved in step 2.1 gives $\int \mathrm{\mid }{g}_{jk}\mathrm{\mid }d\nu \le 2R^{2}m<\mathrm{\infty }$, so [L4] makes $g_{jk}$ integrable and gives $\mathrm{\mid }\int g_{jk}d\nu \mathrm{\mid }\le 2R^{2}m$. Likewise [L4] and step 2.1 give $\mathrm{\mid }\int {\rho }_{j}d\nu \mathrm{\mid }\le Rm$. If $m>0$, $\nu$ is a mass distribution and these are its moment and inertia components by Mass distributions on ${\mathbb{R}}^3$: total mass, first moment and inertia matrix. If $m=0$, the same bounds force every displayed integral to be zero; this case is excluded from the definition's mass-distribution terminology. Thus clause 2 holds for every measure in the hypotheses.