Strict positive covariance with an increasing payoff ($\mathrm{Cov}(g(Z),Z)>0$)

Let $Z$ be a nondegenerate real random variable and $g$ a strictly increasing function. Assume $Z$, $g(Z)$ and $Zg(Z)$ are integrable. For an independent copy $Z^{\prime }$,The product is positive whenever $Z\ne Z^{\prime }$ and zero otherwise. Nondegeneracy gives positive probability of unequal copies, proving strict positivity. The same argument applies to a conditional distribution when that conditional law is nondegenerate and has the required integrability.

For a market with one trading period, suppose an increasing terminal payoff is replicated by a risk-free asset holding and a stock holding: $g(Z)=a+bZ$. Under current information $\mathcal{G}$, let $a,b$ be $\mathcal{G}$-measurable and $0<\mathrm{Var}(Z\mid \mathcal{G})<\infty$. ThenThis is a conditional one-period statement. General dynamic attainability alone does not specify the affine representation used in this calculation.