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\end{equation}  is valid for any $\epsilon > 0$. Then, we split the range of the functions $Q(z,a)$, $a \in \Lambda$ using a finite sequence of numbers $\beta_1, beta_2, \cdots \beta_n$ such that  \begin{equation}  |\beta_{i+1} - \beta_i | > \frac{\epsilon}{2}, \:\:\: \beta_1=\beta, \beta_n = B B.  \end{equation}  We denote as $T_k$ the event   \begin{equation}  \inf_{a \in \Lambda(\beta_k)} \int Q(z,a)dF(z) - \inf_{a \in \Lambda(\beta_k)} \frac{1}{l} \sum_{i=1}^{l} Q(z_i,a) > \frac{\epsilon}{2}  \end{equation}