 Capped claim moments

For a positive claim and $r>0$, the tail integral formula for moments gives the displayed expression. In particular the first moment is ${\int }_0^M\overline{F}(x)dx$ and the raw second moment is $2{\int }_0^{M}x\overline{F}(x)dx$. If the claim law has a density, capping adds an atom of a measure at $M$ with mass $\mathbb{P}(X\ge M)$.