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hold. \Box  Convergence in (\ref{eq:strict_consistency}) always implies convergence in (\ref{eq:cons_lemma}) but not vice versa.  \subsection{Key Theorem}  The relationship for any $\epsilon > 0$  \begin{equation}  P\bigg\{\sup_{a\in \Lambda} \big |\int Q(z,a)dF(z) - \frac{1}{l}\sum_{i=1}^{l}Q(z_i,a) \big | > \epsilon \bigg \} \xrightarrow[l \rightarrow \infty]{} 0  \end{equation}