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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.

Let $A_{\delta }={\sup }_{K\ge 0}{K}^{\delta }C(K)<\infty$. The call-price decay and moment threshold follows by splitting the preceding integral at one. Since $(S-K{)}_{+}\le S$, for $0<\epsilon <\delta$,For $K\ge 1$, the decay bound givesThe power payoff static call representation therefore yieldsThe $\epsilon =0$ case is the given finite first moment. The strict endpoint matters: a Pareto distribution with $\mathbb{P}(S>x)=x^{-(1+\delta )}$ for $x\ge 1$ has $C(K)=K^{-\delta }/\delta$ for $K\ge 1$, but its moment of order $1+\delta$ is infinite. Thus the stated decay condition does not generally imply the endpoint moment.