Highest density region ($S_c=\{\theta :\pi (\theta )\ge c\}$)

For a probability density $\pi$ and threshold $c>0$, this region includes precisely the points at or above the threshold. It is a Bayes act for loss $c|S|-1_S(\theta )$, since expected loss is ${\int }_S(c-\pi (\theta ))d\theta$. The minimal value is $-\int (\pi -c{)}_{+}d\theta$; adding or removing threshold-equality points does not change it.