the appropriate homogenized equations.

Discrete approximation of the Young measures\[sub:Discrete-approximation-of\]
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Here we discuss the derivation of the discrete approximation (\[eq:discrete approx of Young measure, intro\]), from which equations (\[eq:transport multiband-multigroup, intro\]) and (\[eq:formula for psi\_o, discretized Young measures\]) follow from the homogenized equations (\[eq:transport equation on extended phase space, intro\]) and (\[eq:psi\_0 as weighting against Young measure, intro\]). As we will see, the number of bands $\left[\sigma_{j},\sigma_{j+1}\right]$ needed in the approximation to determined by the particular function $F\left(\xi\right)$ used in the fundamental representation (\[eq:Young measure, key rep-1\]) for the Young measure; in our applications, $F\left(\xi\right)$ is very smooth, and a small number of bands are required, independent of the scale $\varepsilon$ at which $\sigma_{\varepsilon}\left(E\right)$ varies in energy.

We will construct a discrete approximation to the Young measure via the theory developed in [@BALL:1989]; for notational simplicity, in this section we drop the temperature dependence in the notation. In particular, the measure $\lambda_{E}$ is entirely determined by its action on continuous functions $f$ via $$\left\langle \lambda_{E},f\right\rangle =\lim_{\Delta E\rightarrow0}\lim_{\varepsilon\rightarrow0}\frac{1}{\Delta E}\int_{E-\Delta E/2}^{E+\Delta E/2}f\left(E',\sigma_{\varepsilon}\left(E'\right)\right)dE'.$$ Note the order of the limits intuitively corresponds to choosing a scale $\delta$ that is large relative to the microscopic behavior but small relative to the macroscopic behavior, $0<\varepsilon\ll\Delta 