discrete approximation of the Young measure $\lambda_{E}^{T}$, we approximate for each energy group $\left[E_{i},E_{i+1}\right]$, $$\mu_{i}^{T}\left(\sigma\right)\approx\sum_{j=1}^{m}p_{i,j}\delta\left(\sigma-\sigma_{i,j}\right),\label{eq:discrete rep of Young measure, example}$$ where $p_{i,j}$ is proportional to the probability that $\sigma_{i,j}\leq\sigma_{\varepsilon}\left(E\right)\leq\sigma_{i,j+1}$ for $E_{i}\leq E\leq E_{i+1}$; $p_{i,j}$ is computed by uniformly sampling random numbers from $\left[E_{i},E_{i+1}\right]$ (the number of samples is chosen to be much larger than the number of energy values needed to resolve $\sigma_{\varepsilon}\left(E\right)$ in $\left[E_{i},E_{i+1}\right]$), counting how many times $ $$\sigma_{i,j}\leq\sigma_{\varepsilon}\left(E\right)\leq\sigma_{i,j+1}$ for each band $\left[\sigma_{i,j},\sigma_{i,j+1}\right]$, and normalizing by the total number of samples. We emphasize that, although evaluating $p_{i,j}$ scales linearly in $\varepsilon^{-1}$, this is a pre-computation and need only be done once; this pre-computation is analogous to computing Planck-weighted or Rosseland-weighted opacities.

Using the discrete representation (\[eq:discrete rep of Young measure, example\]), we have that for $\mu>0$, $$\begin{aligned}
\lim_{\varepsilon\rightarrow0}\int_{E_{i}}^{E_{i+1}}\psi_{\varepsilon}\left(x,\mu,E\right)dE & = & \int_{E_{i}}^{E_{i+1}}S\left(E\right)\left(\int_{0}^{\mathbb{R}}\frac{\left(1-e^{-x\left(\sigma/\mu\right)}\right)}{\sigma}d\lambda_{E}^{T}\left(\sigma\right)\right)dE\nonumber \\
 & \approx & \int_{E_{i}}^{E_{i+1}}S\left(E\right)\left(\sum_{j=1}^{m}p_{i,j}\frac{\left(1-e^{-x\left(\sigma_{i,j}/\mu\right)}\right)}{\sigma_{i,j}}\right)dE\nonumber \\
 & = & S_{i}\sum_{j=1}^{m}p_{i,j}\frac{\left(1-e^{-x\left(\sigma_{i,j}/\mu\right)}\right)}{\sigma_{i,j}},\label{eq:mu > 0, homogenized}\end{aligned}$$ where $$S_{i}=\int_{E_{i}}^{E_{i+1}}S\left(E\right)dE.$$ Similarly, for $\mu<0$, $$\lim_{\varepsilon\rightarrow0}\int_{E_{i}}^{E_{i+1}}\psi_{\varepsilon}\left(s,E\right)dE\approx S_{i}\sum_{j=1}^{m}p_{i,j}\left(1-e^{-\left(x-1\right)\left(\sigma_{i,j}/\mu\right)}\right).\label{eq:mu < 0, homogenized}$$

In Sections \[sub:Example-1:-the\] and \[sub:Example-2:-iron\], we compare the discrete approximation, (\[eq:mu &gt; 0, homogenized\])-(\[eq:mu &lt; 0, homogenized\]), to the exact solution integrated over $\left[E_{i},E_{i+1}\right]$, $$\int_{E_{i}}^{E_{i+1}}\psi_{\varepsilon}\left(x,\mu,E\right)dE=\int_{E_{i}}^{E_{i+1}}S\left(E\right)\frac{\left(1-e^{-\left(\sigma_{\varepsilon}\left(E\right)/\mu\right)x}\right)}{\sigma_{\varepsilon}\left(E\right)}dE,$$ for both the Elsasser band (\[eq:Elsasser band model, intro\]) and for iron opacity as generated via the NJoy 