 Solution

The conditional class probability is $p_k(x)=\mathbb{P}(Y=k\mid X=x)$. A Bayes classifier choosesand its Bayes risk isA sequence of classifiers is consistent when $R(h_n)\to R_{\mathrm{Bayes}}$ as $n\to \infty$, with convergence interpreted in probability or in expectation according to whether the training sample is conditioned upon.