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Maximize $px-x^2/2$ over $|x|\le 1$. The maximizer is $x=\max (-1,\min (p,1))$, giving the displayed Huber loss. The indicator functional of a constraint set is zero on the interval and infinity outside; it is not the zero-one indicator used in measure theory.

Lebesgue integration constructs an integral from the measure of level sets or from approximation by simple functions. It is linear on integrable functions and is stable under the principal convergence theorems of measure theory.