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For integrable real $g$, including points where $g<0$ decreases the integral and including points where $g>0$ increases it. Thus the negative set minimizes the integral, up to null sets and arbitrary choices where $g=0$. This elementary signed-integral fact derives the Bayes classifier and cost-sensitive Bayes classifier.

With zero cost for correct assignments, the total expected misclassification cost isThe part depending on $R_1$ is the integral of $c(1\mid 2){\pi }_2{f}_2-c(2\mid 1){\pi }_1{f}_1$. Therefore the cost-sensitive Bayes classifier isFor positive costs and weighted densities, the log posterior-odds score from part (ii) now has threshold $\log (c(1\mid 2)/c(2\mid 1))$. Larger cost of sending class one to class two expands the region assigned to class one. The weighted-density comparison also handles zero costs without taking an undefined logarithm.