 Solution

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.