= 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 when
$$
\pi_1f_1(x)>\pi_0f_0(x),
$$
and 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 set
$$
B=\{x:\pi_1f_1(x)=\pi_0f_0(x)\}
=\{x:D_{\pi_0}(x)=0\}.
$$
Because 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.
Back to article page
