Inequality for classifiers

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This is from "A Probabilistic Theory of Pattern Recognition". Namely, why is this inequality true?


The inequality:

Inequality


I have included the relevant information below.

Page 69:

Page 69


Page 70:

Page 70


Page 65, AKA the lemmas:

Page 65, AKA the lemmas


This is where $\Psi$ is defined:

This is where Psi is defined.