= Positive-weight expectation cone
{title2=$C_Y=\{\mathbb E[ZY]:0<Z\in L^\infty\}$}
For an <integrable> finite-dimensional <random vector> $Y$, let $L=\{h:h\cdot Y=0\text{ almost surely}\}^\perp$. The displayed <convex cone> is open relative to $L$. Given a bounded strictly positive $Z$, the perturbation $Z_\varepsilon=Z(1+\varepsilon\cdot Y/(1+\|Y\|))$, $\|\varepsilon\|<1$, changes its weighted <expectation> by $A_Z\varepsilon$, where
$$
A_Z=\mathbb E\frac{ZYY^T}{1+\|Y\|}.
$$
For $0\ne v\in L$, $v^TA_Zv>0$: otherwise $v\cdot Y=0$ <almost surely>. Thus $A_Z$ is invertible on $L$, and these perturbations give a neighborhood of every point of the cone. The <hyperplane separation theorem> consequently gives the alternative: either zero is a strictly positively weighted <expectation>, or a nonzero direction in $L$ has a nonnegative <dot product> with $Y$ <almost surely> and a positive <dot product> with positive <probability>. To obtain the latter assertion, test the separating direction with $Z=\mathbf1_{\{h\cdot Y<0\}}+\varepsilon$ and let $\varepsilon\downarrow0$.
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