Cutoff elastic-reflection collision measure and gain

Assume the Axiom of Countable Choice, written ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Under this assumption Lebesgue measure is the completion of Borel measure ($\mathcal{L}({\mathbb{R}}^n)$ is exactly the completion of the restriction of ${\lambda }_n$ to the Borel sets), and every Lebesgue-measurable integrable class has a Borel representative (A function measurable for a completion is almost everywhere equal to one measurable for the original sigma-algebra). Write ${\lambda }_3$ for Lebesgue measure on ${\mathbb{R}}^3$ (Lebesgue measurable sets, the family $\mathcal{L}({\mathbb{R}}^n)$, and the restricted set function ${\lambda }_n$). Fix a positive reference speed $v_0>0$, and put $w_0(v)=1+\mathrm{\mid }v{\mathrm{\mid }}^2\mathrm{/}{v}_0^2$. Let $X$ be the real vector space of equivalence classes modulo ${\lambda }_3$-almost-everywhere equality of measurable functions $f:{\mathbb{R}}^3\to \mathbb{R}$ with

$\mathrm{\parallel }f{\mathrm{\parallel }}_{1,w_0}:={\int }_{{\mathbb{R}}^3}{w}_0(v)\mathrm{\mid }f(v)\mathrm{\mid }d{\lambda }_3(v)<\mathrm{\infty }\mathrm{.}$

Let $S^2=\{n\in {\mathbb{R}}^3:\mathrm{\mid }n\mathrm{\mid }=1\}$. For a Borel set $E\subseteq S^2$, define the normalized finite angular measure

$\sigma (E):=\frac{1}{4\pi }{\int }_0^{\pi }{\int }_0^{2\pi }{\mathbf{1}}_E(\sin \theta \cos \varphi ,\sin \theta \sin \varphi ,\cos \theta )\sin \theta d\varphi d\theta \mathrm{.}$

For $v,w\in {\mathbb{R}}^3$ and $n\in S^2$, define the postcollision velocities

$v^{\mathrm{\prime }}=v-((v-w)\cdot n)n,w^{\mathrm{\prime }}=w+((v-w)\cdot n)n\mathrm{.}$

The collision-parameter measure is $d\mu =d{\lambda }_3(v)d{\lambda }_3(w)d\sigma (n)$ on ${\mathbb{R}}^3\times {\mathbb{R}}^3\times S^2$. Its angular factor is finite and normalized; there is no singular angular weight. For $f,g\in X$, choose Borel representatives and define the gain $\mathcal{G}(f,g)$ by the almost-everywhere integral

$\mathcal{G}(f,g)(v):={\int }_{{\mathbb{R}}^3\times S^2}f(v^{\mathrm{\prime }})g(w^{\mathrm{\prime }})d{\lambda }_3(w)d\sigma (n)\mathrm{.}$

The value on any exceptional null set is immaterial because the output is an equivalence class. The next item proves that this definition is independent of the representatives, maps $X\times X$ to $X$, and has the stated collision and moment identities.