defined via (\[eq:definition of p\_j(E)\]).

The function $p_{j}\left(E\right)$ give the probability that $\sigma_{j}\leq\sigma_{\varepsilon}\left(E'\right)\leq\sigma_{j+1}$ for $E'\in\left[E-\Delta E/2,E+\Delta E/2\right]$, where $0<\varepsilon\ll\Delta E$; in particular, $p_{j}\left(E\right)$ may be computed by uniformly sampling $E'\in\left[E-\Delta E/2,E+\Delta E/2\right]$ and counting how many times $\sigma_{j}\leq\sigma_{\varepsilon}\left(E'\right)\leq\sigma_{j+1}$ for each band $\left[\sigma_{j},\sigma_{j+1}\right]$. We remark that $p_{j}\left(E\right)$ can be precomputed on a fine energy grid, and evaluated at other energy points via interpolation.

Derivation of the discrete homogenized system\[sub:Derivation-of-discrete\]
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To derive (\[eq:transport multiband-multigroup, intro\]), evaluate (\[eq:transport equation on extended phase space, intro\]) at $\xi=\sigma_{i,j}$ and integrate in $E$ over $\left[E_{i},E_{i+1}\right]$. Now, approximate the integral $$\int_{E_{i}}^{E_{i+1}}\chi\left(\sigma_{i,j},E,T\left(\mathbf{x}\right)\right)\Psi\left(\mathbf{x},\boldsymbol{\Omega},E,\sigma_{i,j}\right)dE$$ by $$\chi\left(\sigma_{i,j},E_{i},T\left(\mathbf{x}\right)\right)\int_{E_{i}}^{E_{i+1}}\Psi\left(\mathbf{x},\boldsymbol{\Omega},E,\sigma_{i,j}\right)\Psi dE=\kappa_{ij}\left(T\left(\mathbf{x}\right)\right)\Psi_{ij}\left(\mathbf{x},\boldsymbol{\Omega},E\right).$$ This step is accurate since, by assumption, $\chi\left(\xi,E,T\left(\mathbf{x}\right)\right)$ smoothly varies in energy $E$. We perform a similar calculation for the integral of $\chi\left(\sigma_{i,j},E,T\left(\mathbf{x}\right)\right)S\left(T\left(\mathbf{x}\right),E\right)$ over $\left[E_{i},E_{i+1}\right]$, and obtain the multigroup-type system (\[eq:transport multiband-multigroup, intro\]).

To derive (\[eq:formula for psi\_o, discretized Young measures\]), we use the discrete approximation (\[eq:discrete approx of Young measure, intro\]) in (\[eq:psi\_0 as weighting against Young measure, intro\]), $$\begin{aligned}
\psi_{0}\left(\mathbf{x},\boldsymbol{\Omega},E\right) & = & \int_{0}^{\infty}\Psi\left(\mathbf{x},\boldsymbol{\Omega},E,\xi\right)\lambda_{E}\left(d\xi\right)\\
 & \approx & \int_{0}^{\infty}\Psi\left(\mathbf{x},\boldsymbol{\Omega},E,\xi\right)\left(\sum_{j=1}^{m}p_{i,j}\delta\left(\xi-\sigma_{i,j}\right)\right)d\xi\\
 & = & \sum_{j=1}^{m}p_{i,j}\Psi\left(\mathbf{x},\boldsymbol{\Omega},E,\sigma_{i,j}\right).\end{aligned}$$ Finally, integrating both sides of $$\psi_{0}\left(\mathbf{x},\boldsymbol{\Omega},E\right)=\sum_{j=1}^{m}p_{i,j}\Psi\left(\mathbf{x},\boldsymbol{\Omega},E,\sigma_{i,j}\right)$$ over $\left[E_{i},E_{i+1}\right]$, we obtain (\[eq:formula for psi\_o, discretized Young measures\]).

Relaxation of the correlated opacity assumption\[sub:Relaxation-of-the\]
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Our derivation of the homogenized 