on angle, i.e. $\Sigma^{s}=\Sigma^{s}\left(\mathbf{x},\boldsymbol{\Omega}\cdot\boldsymbol{\Omega}'\right)$. Consider the solution $\Psi\left(\mathbf{x},\boldsymbol{\Omega},E,\xi\right)$ of (\[eq:transport equation on extended phase space, intro\]). Then $\Psi\left(\mathbf{x},\boldsymbol{\Omega},E,\sigma_{\varepsilon}\left(E\right)\right)$ satisfies the transport equation (\[eq:transport equation, intro\]). Therefore, since (\[eq:transport equation, intro\]) has a unique solution, $\psi_{\varepsilon}\left(\mathbf{x},\boldsymbol{\Omega},E\right)=\Psi\left(\mathbf{x},\boldsymbol{\Omega},E,\sigma_{\varepsilon}\left(E\right)\right)$. It follows from (\[eq:Young measure, key rep-1\]) that, for any $0<E_{0}<E_{1}$, $$\begin{aligned}
\lim_{\varepsilon\rightarrow0}\int_{E_{0}}^{E_{1}}\psi_{\varepsilon}\left(\mathbf{x},\boldsymbol{\Omega},E\right)dE & = & \lim_{\varepsilon\rightarrow0}\int_{E_{0}}^{E_{1}}\Psi\left(\mathbf{x},\boldsymbol{\Omega},E,\sigma_{\varepsilon}\left(E\right)\right)dE.\\
 & = & \int_{E_{0}}^{E_{1}}\int_{0}^{\infty}\Psi\left(\mathbf{x},\boldsymbol{\Omega},E,\xi\right)d\lambda_{E}\left(\xi\right)dE\end{aligned}$$ Since $E_{0}$ and $E_{1}$ are arbitrary, we finally conclude that $$\psi_{0}\left(\mathbf{x},\boldsymbol{\Omega},E\right)=\int_{0}^{\infty}\Psi\left(\mathbf{x},\boldsymbol{\Omega},E,\xi\right)\lambda_{E}\left(d\xi\right).$$

Now suppose that the scattering kernel $\Sigma^{s}=\Sigma^{s}\left(\mathbf{x},E,E',\boldsymbol{\Omega}\cdot\boldsymbol{\Omega}'\right)$ depends on both energy and angle. We again argue that (\[eq:transport equation on extended phase space, intro\]) and (\[eq:psi\_0 as weighting against Young measure, intro\]) are the appropriate homogenized equations. To do so, define $\tilde{\psi}_{\varepsilon}\left(\mathbf{x},\boldsymbol{\Omega},E\right)\equiv\Psi\left(\mathbf{x},\boldsymbol{\Omega},E,\sigma_{\varepsilon}\left(E\right)\right)$. Then $$\boldsymbol{\Omega}\cdot\nabla_{\mathbf{x}}\tilde{\psi}_{\varepsilon}+\sigma_{\varepsilon}\tilde{\psi}_{\varepsilon}=\sigma_{\varepsilon}^{a}S+Q_{0}+\int_{\mathbb{S}^{2}}\left(\int_{0}^{\infty}\Sigma^{s}\left(\mathbf{x},E,E',\boldsymbol{\Omega}\cdot\boldsymbol{\Omega}'\right)\tilde{\psi}_{\varepsilon}dE'd\boldsymbol{\Omega}'+\mathcal{E}_{\varepsilon}\right)d\boldsymbol{\Omega}',$$ where the residual term $\mathcal{E}_{\varepsilon}$ is given by $$\mathcal{E}_{\varepsilon}\left(\mathbf{x},\boldsymbol{\Omega}',E\right)=\int_{0}^{\infty}\Sigma^{s}\left(\mathbf{x},E,E',\boldsymbol{\Omega}\cdot\boldsymbol{\Omega}'\right)\left(\tilde{\psi}_{\varepsilon}\left(\mathbf{x},\boldsymbol{\Omega}',E'\right)-\int_{0}^{\infty}\Psi\left(\mathbf{x},\boldsymbol{\Omega}',E',\xi\right)\lambda_{E'}\left(d\xi\right)\right)dE'.$$ From the property (\[eq:Young measure, key rep-1\]) and $\tilde{\psi}_{\varepsilon}\left(\mathbf{x},\boldsymbol{\Omega},E'\right)=\Psi\left(\mathbf{x},\boldsymbol{\Omega},E',\sigma_{\varepsilon}\left(E'\right)\right)$, we see that $\mathcal{E}_{\varepsilon}\left(\mathbf{x},\boldsymbol{\Omega},E\right)\rightarrow0$ for $\varepsilon\rightarrow0$. In addition, the energy dependence in $\mathcal{E}_{\varepsilon}\left(\mathbf{x},\boldsymbol{\Omega},E\right)$ is only through $\Sigma^{s}\left(\mathbf{x},E,E',\boldsymbol{\Omega}\cdot\boldsymbol{\Omega}'\right)$, and is therefore slow (i.e., it is independent of the small parameter $\varepsilon$). It follows that for $\varepsilon\ll1$, $$\boldsymbol{\Omega}\cdot\nabla_{\mathbf{x}}\tilde{\psi}_{\varepsilon}+\sigma_{\varepsilon}\tilde{\psi}_{\varepsilon}-\sigma_{\varepsilon}^{a}S+Q_{0}+\int_{0}^{\infty}\Sigma^{s}\left(\mathbf{x},E,E',\boldsymbol{\Omega}\cdot\boldsymbol{\Omega}'\right)\tilde{\psi}_{\varepsilon}dE\approx0.$$ Since the transport equation is well-posed, $\tilde{\psi}_{\varepsilon}\approx\psi_{\varepsilon}$. Finally, invoking (\[eq:Young measure, key rep-1\]) again, $$\lim_{\varepsilon\rightarrow0}\int_{E_{0}}^{E_{1}}\tilde{\psi}_{\varepsilon}\left(\mathbf{x},\boldsymbol{\Omega},E\right)dE=\int_{E_{0}}^{E_{1}}\int_{0}^{\infty}\Psi\left(\mathbf{x},\boldsymbol{\Omega},E,\xi\right)\lambda_{E}\left(d\xi\right)dE,$$ and we see that (\[eq:transport equation on extended phase space, intro\]) and (\[eq:psi\_0 as weighting against Young measure, intro\]) are 