homogenized solution at $\left(\mathbf{x},\boldsymbol{\Omega},E\right)$ in the limit of $\varepsilon\rightarrow0$. The proof of this is a direct application of (\[eq:Young measure, key rep-1\]), and is given in Section \[sec:The-homogenized-transport\]. In the proof, we assume that (\[eq:transport equation on extended phase space, intro\]) has a unique solution that is continuous in its last argument $\xi$.

By discretizing equations (\[eq:transport equation on extended phase space, intro\]) and (\[eq:psi\_0 as weighting against Young measure, intro\]), we obtain an algorithm that is analogous to the multiband method (cf. [@CUL-POM:1980]). In particular, we choose a coarse number $m_{g}$ of energy groups $\left[E_{i},E_{i+1}\right]$ and a coarse number $m_{\sigma}$ opacity bands $\left[\sigma_{j},\sigma_{j+1}\right]$, and use the theory in [@BALL:1989] to construct a discrete approximation $$\lambda_{E}\left(\xi\right)\approx\sum_{j=1}^{m}p_{i,j}\delta\left(\xi-\sigma_{i,j}\right),\,\,\,\, E\in\left[E_{i},E_{i+1}\right],\label{eq:discrete approx of Young measure, intro}$$ where $p_{i,j}$ gives the probability that $\sigma_{j}\leq\sigma_{\varepsilon}\left(E\right)\leq\sigma_{j+1}$ for $E\in\left[E_{i},E_{i+1}\right]$ (see Section \[sub:Discrete-approximation-of\] for more details). Convergence of (\[eq:discrete approx of Young measure, intro\]) to $\lambda_{E}$ is made precise in [@BALL:1989]. The key point of this construction is that, for realistic opacities, the number parameters $\sigma_{i,j}$ and $p_{i,j}$ is typically a small constant independent of the size $\varepsilon$ of the microscale; this is provably so when the opacity is multiscale, e.g. of the form $\sigma_{0}\left(T,E,E/\varepsilon\right)$, where, e.g., $\sigma_{0}\left(T,E,\kappa\right)$ is 