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For an integrable real function $g$, let $R_{-}=\{x:g(x)<0\}$. For every measurable region $R$,The first term is nonnegative because $g\ge 0$ there, and the integral subtracted in the second term is nonpositive. Hence the negative set minimizes the integral. Including or excluding any points where $g=0$ has no effect; changes on null sets do not matter either. This is the minimum-integral decision region principle. Integrability ensures the subtraction is defined, and it holds automatically for the weighted differences of probability density functions used below.

Apply the minimum-integral decision region principle to $g(x)={\pi }_2{f}_2(x)-{\pi }_1{f}_1(x)$. The optimal Bayes classifier chooses class one when ${\pi }_1{f}_1(x)>{\pi }_2{f}_2(x)$, and class two for the reverse inequality. Ties can be assigned either way. An everywhere valid discriminant score isWhere both weighted densities are positive one may equivalently use $\log ({\pi }_1{f}_1(x)/({\pi }_2{f}_2(x)))$ and threshold zero. This chooses the largest posterior class probability, because both posterior probabilities have the same positive mixture-density denominator.