Minimum-integral decision region ($\arg {\min }_R{\int }_{R}g=\{x:g(x)<0\}$)

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.