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This question is closely related to another question I asked recently but is more to the point than that other question.

Let $\mathcal P$ be the set of all probability measures on the Borel algebra of $[0,1]$. A measurable function $S: \mathcal P \times [0,1] \to [-\infty, 0]$ is called a strictly proper scoring rule if $$\int S(P,x)P(dx) > \int S(Q, x)P(dx)$$ holds for all $P \neq Q$ in $\mathcal P$.

Can anyone provide a concrete example of a strictly proper scoring rule?

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