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\textbf{Definition.} ERM principle is consistent for the set of functions $Q(z,a)$, $a \in \Lambda$ and the probability distribution $F(z)$ if the following two sequences converge in probability:  \begin{equation}  R(a_l) \xrightarrow[l \rightarrow \infty]{P} \inf_{\a \inf_{a  \in \Lambda}R(a) \end{equation}