E$. Then $$\left\langle \lambda_{E},f\right\rangle \approx\frac{1}{\Delta E}\int_{E-\Delta E/2}^{E+\Delta E/2}f\left(E,\sigma_{\varepsilon}\left(E'\right)\right)dE',\label{eq:Ball, approximation of Young measure}$$ and this becomes precise by first letting $\varepsilon\rightarrow0$ and then $\Delta E\rightarrow0$. In (\[eq:Ball, approximation of Young measure\]), approximated $f\left(E',\sigma_{\varepsilon}\left(E'\right)\right)\approx f\left(E,\sigma_{\varepsilon}\left(E'\right)\right)$, which is valid over $\left[E-\Delta E/2,E+\Delta E/2\right]$ since the variation of $f$ in its first argument does not depend on the fast scale $\varepsilon$.

Given fixed $\sigma_{1}<\dots\sigma_{j}<\sigma_{j+1}<\ldots$, define the characteristic functions $\zeta_{j}\left(\xi\right)$, $$\zeta_{j}\left(\xi\right)=\begin{cases}
1, & \,\sigma_{j}\leq\xi\leq\sigma_{j+1}\\
0, & \,\text{else}.
\end{cases}.$$ Consider the collection of step functions $$f\left(E,\xi\right)=\sum_{j}f\left(E,\sigma_{j}\right)\zeta_{j}\left(\xi\right).\label{eq:step function}$$ Although $\zeta_{j}$ are not continuous, any continuous function can be approximated by such functions. Now, for a general step function (\[eq:step function\]) and using (\[eq:Ball, approximation of Young measure\]), $$\begin{aligned}
\left\langle \lambda_{E},f\right\rangle  & \approx & \frac{1}{\Delta E}\int_{E-\Delta E/2}^{E+\Delta E/2}\left(\sum_{j}f\left(E',\sigma_{j}\right)\zeta_{j}\left(\sigma_{\varepsilon}\left(E'\right)\right)\right)dE'\\
 & \approx & \sum_{j}f\left(E,\sigma_{j}\right)\left(\frac{1}{\Delta E}\int_{E-\Delta E/2}^{E+\Delta E/2}\zeta_{j}\left(\sigma_{\varepsilon}\left(E'\right)\right)dE'\right)\\
 & = & \sum_{j}f\left(E,\sigma_{j}\right)p_{j}\left(E\right).\end{aligned}$$ In the last equality, the probability $p_{j}\left(E\right)$ is given by $$p_{j}\left(E\right)=\frac{\lambda\left(E'\in\left[E-\Delta E/2,E+\Delta E/2\right]\mid\sigma_{j}\leq\sigma_{\varepsilon}\left(E'\right)\leq\sigma_{j+1}\right)}{\Delta E},\label{eq:definition of p_j(E)}$$ with $\lambda$ in (\[eq:definition of p\_j(E)\]) denoting the Lebesque measure. Approximating a general continuous function $f\left(\xi\right)$ (defined for $\xi\geq0$) by a step function, we have that $$\begin{aligned}
\left\langle \lambda_{E},f\right\rangle  & \approx & \sum_{j}p_{j}\left(E\right)f\left(E,\sigma_{j}\right)\\
 & = & \int f\left(E,\xi\right)\left(\sum_{j}p_{j}\left(E\right)\delta\left(\xi-\sigma_{j}\right)\right)d\xi.\end{aligned}$$ Thus, a discrete approximation to the Young measure is given by $$\lambda_{E}\left(\xi\right)\approx\sum_{j}p_{j}\left(E\right)\delta\left(\xi-\sigma_{j}\right),$$ where $p_{j}\left(E\right)$ is 