Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 39 / 2 / a / Solution

For a deterministic $s>0$, the positive part $(s-K{)}_{+}$ vanishes for $K\ge s$, so direct integration givesAt $s=0$ both sides vanish. Applying this pointwise to the nonnegative random variable proves the power payoff static call representationBy Tonelli theorem, the corresponding moment identity isincluding the possibility that both sides are infinite. No higher-moment assumption is needed to interchange these nonnegative integrals.