= Solution
For an integrable real function $g$, let $R_- =\{x:g(x)<0\}$. For every measurable region $R$,
$$
\int_Rg-\int_{R_-}g
=\int_{R\setminus R_-}g-\int_{R_-\setminus R}g\ge0.
$$
The first term is nonnegative because $g\ge0$ there, and the integral subtracted in the second term is nonpositive. Hence \b[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.
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