= Strict positive covariance with an increasing payoff
{title2=$\operatorname{Cov}(g(Z),Z)>0$}
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 $Z'$,
$$
2\operatorname{Cov}(g(Z),Z)=\mathbb E[(g(Z)-g(Z'))(Z-Z')]>0.
$$
The product is positive whenever $Z\ne Z'$ 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 $0<\operatorname{Var}(Z\mid\mathcal G)<\infty$. Then
$$
b=\frac{\operatorname{Cov}(g(Z),Z\mid\mathcal G)}{\operatorname{Var}(Z\mid\mathcal G)}>0.
$$
This is a conditional one-period statement. General dynamic attainability alone does not specify the affine representation used in this calculation.
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