discrete probabilities $p_{j,j'}\left(E,T\right)$ by uniformly sampling energy values $E'$ in $\left[E-\Delta E/2,E+\Delta E/2\right]$ and counting the number of samples for which $\sigma_{j}\left(E,T\right)\leq\sigma_{\varepsilon}\left(E',T\right)\leq\sigma_{j+1}\left(E,T\right)$ and $\sigma_{j'}\left(E,T_{0}\right)\leq\sigma_{\varepsilon}\left(E',T_{0}\right)\leq\sigma_{j'+1}\left(E,T_{0}\right)$.

We approximate the joint Young measure by $$\lambda_{E,T}\left(\xi_{1},\xi_{2}\right)\approx\sum_{j,j'}p_{j,j'}\left(E,T\right)\delta\left(\xi_{1}-\sigma_{j}\left(E,T_{0}\right)\right)\delta\left(\xi_{2}-\sigma_{j'}\left(E,T\right)\right).$$ Then from (\[eq: chi, conditional expected value\]), $$\begin{aligned}
\chi\left(x,E,T\right) & \approx & \int_{0}^{\infty}\xi_{2}\left(\sum_{j,j'}p_{j,j'}\left(E,T\right)\delta\left(x-\sigma_{j}\left(E,T_{0}\right)\right)\delta\left(\xi_{2}-\sigma_{j'}\left(E,T\right)\right)\right)d\xi_{2}\\
 & = & \sum_{j,j'}\sigma_{j'}\left(E,T\right)p_{j,j'}\left(E,T\right)\delta\left(x-\sigma_{j}\left(E,T_{0}\right)\right).\end{aligned}$$ Integrating $x$ from $\sigma_{j}\left(E,T_{0}\right)$ to $\sigma_{j+1}\left(E,T_{0}\right)$, $$\int_{\sigma_{j}\left(E,T_{0}\right)}^{\sigma_{j+1}\left(E,T_{0}\right)}\chi\left(x,E,T\right)dx\approx\sum_{j'}\sigma_{j'}\left(E,T\right)p_{j,j'}\left(E,T\right).$$

To summarize: in the discrete version of the homogenized system (\[eq:transport multiband-multigroup, intro\]), we take $$\kappa_{ij}\left(T\right)=\chi\left(\sigma_{i,j},E_{i},T\right)\approx\sum_{j'}p_{j,j'}\left(E_{i},T\right)\sigma_{j'}\left(E_{i},T\right),\label{eq:kappa_=00007Bi,j=00007D}$$ where $E_{i}$ denotes the left end point of the $i$th coarse group, the temperature-dependent probabilities $p_{j,k}\left(E_{i},T\right)$ are defined by (\[eq:p\_=00007Bi,j=00007D(E,T)\]), and the temperature-dependent bands $\sigma_{k}\left(E_{i},T\right)$ are, e.g., logarithmically spaced between $\min_{E\in\left[E_{i},E_{i+1}\right]}\sigma_{\varepsilon}\left(E,T\right)$ and $\max_{E\in\left[E_{i},E_{i+1}\right]}\sigma_{\varepsilon}\left(E,T\right)$. Notice that the probabilities $p_{j,j'}\left(E,T\right)$ may be pre-computed on a fine energy and temperature grid and evaluated at arbitrary energy and temperature values via interpolation.

Numerical Examples\[sec:Examples\]
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We apply this methodology to three examples. We first consider in Section \[sub:Example-1:-the\] the radiative transfer equation at constant temperature and using the Elsasser band opacity (\[eq:Elsasser band model, intro\]), where we take the line spacing $\varepsilon=10^{-4}$; this example is also considered in [@POMRAN:1971]. For this simple but instructive example, we can compute the Young measure analytically and compare it to its discrete approximation.

In our second 