 Solution

Condition on the observed value $X=x$. If a classifier reports class zero, its conditional probability of error is $\mathbb{P}(Y=1\mid X=x)$; if it reports class one, the conditional error is $\mathbb{P}(Y=0\mid X=x)$. Therefore the smaller conditional error is attained by class one exactly whenand by class zero exactly when the reverse strict inequality holds. Integrating these pointwise conditional errors proves that ${\delta }_{{\pi }_0}$ is a Bayes classifier.

The only possible nonuniqueness lies on the tie setBecause the two Gaussian class distributions are distinct, $D_{{\pi }_0}$ is not identically zero. It is a nonzero polynomial of degree at most two, so its zero set has Lebesgue measure zero. Both Gaussian laws have densities with respect to Lebesgue measure, hence assign probability zero to $B$. Every Bayes rule must therefore agree with ${\delta }_{{\pi }_0}$ almost surely. This proves the uniqueness of a Bayes classifier up to the standard null-set equivalence of decision rules.