Radon\] For most purposes of this paper, we could have chosen, instead of the particular measure $\mu_S$, any positive Radon measure $\rho$ on (Borel sets of) the unit sphere $S$, as long as relevant additional assumptions are supposed, for example that $\Gamma_0$ has measure zero with respect to $\sigma\otimes \rho\otimes {\mathcal{L}}^1$, i.e. $\int_{\Gamma} \chi_{\Gamma_0}(y,\omega,E)d\sigma(y)d\rho(\omega)dE=0$. Note that it follows from this assumption that $N_0$ has measure zero as well.

Especially, the main existence and uniqueness results for the solutions of the CSDA Boltzmann transport problem, as given in Sections \[single-eq\] and \[cosyst\] (Theorems \[csdath3\], \[coth2-d\], \[evoth1\], \[cosystth2\], \[m-d-j-co1\], \[coupthev\], along with their corollaries), as well as the non-negativity result of Subsection \[possol\], remain essentially true even with this more general choice of measure on $S$.

Below the maps $t:G\times S\to [0,\infty]$ and $\tau_{\pm}:\partial G\times S\to [0,\infty]$, will often be interpret as maps $t:G\times S\times I\to [0,\infty]$ and $\tau_{\pm}:\Gamma\to [0,\infty]$, by dropping off the energy variable, e.g. $t(x,\omega,E)=t(x,\omega)$.

\[le:esccont\] Let $(y_0,\omega_0)\in \partial G\times S$ be such that there exists a bounded open cone $C:=\{x\in{\mathbb{R}}^3\ |\ \Big|{x\over{{\left\Vert x\right\Vert}}}-{a\over{{\left\Vert a\right\Vert}}}\Big|<r, 0<{\left\Vert x\right\Vert}<r\}\subset{\mathbb{R}}^3,\ a\not=0$ containing $-\omega_0$ such that for some $\lambda_0>0$ one has $y_0+\lambda_0 C\subset {\mathbb{R}}^3\backslash G$. Then we have the following:

\(i) The map $t(\cdot,\cdot)$ is continuous at any point $(x_0,\omega_0)\in G\times S$ such that $y_0=x_0-t(x_0,\omega_0)\omega_0$.

\(ii) Letting $y_+ = y_0 + \tau_-(y_0,\omega_0)\omega_0$, one has the following limits $$& \lim_{(x,\omega)\to (y_0,\omega_0)} t(x,\omega)=0, \\
& \lim_{(x,\omega)\to (y_+,\omega_0)} t(x,\omega)=\tau_-(y_0,\omega_0),$$ where $(x,\omega)\in G\times S$ when taking the limits.

By using c